def render_glnQ_group(args):
info = {}
if 'label' in args:
label = clean_input(args['label'])
info = db.gps_qrep.lucky({'label': label})
info['dispmat'] = dispmat
info['groupname'] = '${}$'.format(group_names_pretty(info['group']))
info['groupknowl'] = abstract_group_display_knowl(
info['group'], info['groupname'])
title = r'$\GL(' + str(info['dim']) + r',\Q)$ subgroup ' + label
prop = [('Label', '%s' % label), ('Order', r'\(%s\)' % info['order']),
('Dimension', '%s' % info['dim'])]
bread = get_bread([(label, )])
# downloads = [('Code to Magma', url_for(".hgcwa_code_download", label=label, download_type='magma')),
# ('Code to Gap', url_for(".hgcwa_code_download", label=label, download_type='gap'))]
return render_template(
"glnQ-show-group.html",
title=title,
bread=bread,
info=info,
properties=prop,
#friends=friends,
learnmore=learnmore_list(),
#downloads=downloads,
credit=credit_string)
def hgm_jump(info):
label = clean_input(info['jump'])
if HGM_LABEL_RE.match(label):
return redirect(url_for_label(normalize_motive(label)), 301)
if HGM_FAMILY_LABEL_RE.match(label):
return redirect(url_for_label(normalize_family(label)), 301)
flash_error('%s is not a valid label for a hypergeometric motive or family of hypergeometric motives', label)
return redirect(url_for(".index"))
def hgm_jump(info):
label = clean_input(info['jump_to'])
if HGM_LABEL_RE.match(label):
return render_hgm_webpage(normalize_motive(label))
if HGM_FAMILY_LABEL_RE.match(label):
return render_hgm_family_webpage(normalize_family(label))
flash_error('%s is not a valid label for a hypergeometric motive or family of hypergeometric motives', label)
return redirect(url_for(".index"))
def render_group_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
label = label.replace('t', 'T')
C = base.getDBConnection()
data = C.transitivegroups.groups.find_one({'label': label})
if data is None:
bread = get_bread([("Search error", url_for('.search'))])
info['err'] = "Group " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
title = 'Galois Group:' + label
n = data['n']
t = data['t']
data['yesno'] = yesno
order = data['order']
data['orderfac'] = latex(ZZ(order).factor())
orderfac = latex(ZZ(order).factor())
data['ordermsg'] = "$%s=%s$" % (order, latex(orderfac))
if ZZ(order) == 1:
data['ordermsg'] = "$1$"
if ZZ(order).is_prime():
data['ordermsg'] = "$%s$ (is prime)" % order
pgroup = len(ZZ(order).prime_factors()) < 2
if n == 1:
G = gap.SmallGroup(n, t)
else:
G = gap.TransitiveGroup(n, t)
if ZZ(order) < ZZ('10000000000'):
ctable = chartable(n, t)
else:
ctable = 'Group too large'
data['gens'] = generators(n, t)
if n == 1 and t == 1:
data['gens'] = 'None needed'
data['chartable'] = ctable
data['parity'] = "$%s$" % data['parity']
data['cclasses'] = conjclasses(G, n)
data['subinfo'] = subfield_display(C, n, data['subs'])
data['resolve'] = resolve_display(C, data['resolve'])
# if len(data['resolve']) == 0: data['resolve'] = 'None'
data['otherreps'] = otherrep_display(n, t, C, data['repns'])
prop2 = [
('Order:', '\(%s\)' % order),
('n:', '\(%s\)' % data['n']),
('Cyclic:', yesno(data['cyc'])),
('Abelian:', yesno(data['ab'])),
('Solvable:', yesno(data['solv'])),
('Primitive:', yesno(data['prim'])),
('$p$-group:', yesno(pgroup)),
('Name:', group_display_short(n, t, C)),
]
info.update(data)
bread = get_bread([(label, ' ')])
return render_template("gg-show-group.html", credit=GG_credit, title=title, bread=bread, info=info, properties2=prop2)
def by_label(label):
try:
nflabel = nf_string_to_label(clean_input(label))
if label != nflabel:
return redirect(url_for(".by_label", label=nflabel), 301)
return render_field_webpage({'label': nflabel})
except ValueError as err:
flash_error("%s is not a valid input for a <span style='color:black'>label</span>. %s", label, str(err))
return redirect(url_for(".number_field_render_webpage"))
def by_label(label):
try:
nflabel = nf_string_to_label(clean_input(label))
if label != nflabel:
return redirect(url_for(".by_label", label=nflabel), 301)
return render_field_webpage({'label': nflabel})
except ValueError as err:
flash_error("<span style='color:black'>%s</span> us not a valid input for a <span style='color:black'>label</span>. "+ str(err), label)
return redirect(url_for(".number_field_render_webpage"))
def by_label(label):
clean_label = clean_input(label)
if clean_label != label:
return redirect(url_for('.by_label', label=clean_label), 301)
if label == '1.2.1.d1':
return redirect(url_for('.by_label', label='1.2.1.2.1a'), 301)
if label == '1.2.3.c1':
return redirect(url_for('.by_label', label='1.2.3.1.1a'), 301)
return search_by_label(label)
def by_label(label):
clean_label = clean_input(label)
if clean_label != label:
return redirect(url_for('.by_label', label=clean_label), 301)
if label == '1.2.1.d1':
return redirect(url_for('.by_label', label='1.2.1.2.1a'), 301)
if label == '1.2.3.c1':
return redirect(url_for('.by_label', label='1.2.3.1.1a'), 301)
return search_by_label(label)
def render_hgm_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
C = base.getDBConnection()
data = C.hgm.motives.find_one({'label': label})
if data is None:
bread = get_bread([("Search error", url_for('.search'))])
info['err'] = "Motive " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
title = 'Hypergeometric Motive:' + label
A = data['A']
B = data['B']
tn,td = data['t']
t = latex(QQ(str(tn)+'/'+str(td)))
primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71]
locinfo = data['locinfo']
for j in range(len(locinfo)):
locinfo[j] = [primes[j]] + locinfo[j]
locinfo[j][2] = poly_with_factored_coeffs(locinfo[j][2], primes[j])
hodge = data['hodge']
prop2 = [
('Degree', '\(%s\)' % data['degree']),
('Weight', '\(%s\)' % data['weight']),
('Conductor', '\(%s\)' % data['cond']),
]
# Now add factorization of conductor
Cond = ZZ(data['cond'])
if not (Cond.abs().is_prime() or Cond == 1):
data['cond'] = "%s=%s" % (str(Cond), factorint(data['cond']))
info.update({
'A': A,
'B': B,
't': t,
'degree': data['degree'],
'weight': data['weight'],
'sign': data['sign'],
'sig': data['sig'],
'hodge': hodge,
'cond': data['cond'],
'req': data['req'],
'locinfo': locinfo
})
AB_data, t_data = data["label"].split("_t")
#AB_data = data["label"].split("_t")[0]
friends = [("Motive family "+AB_data.replace("_"," "), url_for(".by_family_label", label = AB_data))]
friends.append(('L-function', url_for("l_functions.l_function_hgm_page", label=AB_data, t='t'+t_data)))
# if rffriend != '':
# friends.append(('Discriminant root field', rffriend))
bread = get_bread([(label, ' ')])
return render_template("hgm-show-motive.html", credit=HGM_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends)
def parse_artin_label(label, safe=False):
try:
label = clean_input(label)
if LABEL_RE.match(label):
return label
if OLD_LABEL_RE.match(label):
newlabel = db.artin_old2new_labels.lookup(label)['new']
if newlabel:
return newlabel
except:
if safe:
return ''
raise ValueError
def by_label(label):
if label == "random":
#This version leaves the word 'random' in the URL:
#return render_field_webpage({'label': label})
return redirect(url_for_label(db.nf_fields.random()), 301)
try:
nflabel = nf_string_to_label(clean_input(label))
if label != nflabel:
return redirect(url_for_label(nflabel), 301)
return render_field_webpage({'label': nflabel})
except ValueError as err:
flash_error("%s is not a valid input for a <span style='color:black'>label</span>. %s", label, str(err))
return redirect(url_for(".number_field_render_webpage"))
def render_hgm_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
C = base.getDBConnection()
data = C.hgm.motives.find_one({'label': label})
if data is None:
bread = get_bread([("Search error", url_for('.search'))])
info['err'] = "Motive " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
title = 'Hypergeometric Motive:' + label
A = data['A']
B = data['B']
tn = data['t']['n']
td = data['t']['d']
t = latex(QQ(str(tn)+'/'+str(td)))
primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71]
locinfo = data['locinfo']
for j in range(len(locinfo)):
locinfo[j] = [primes[j]] + locinfo[j]
locinfo[j][2] = poly_with_factored_coeffs(locinfo[j][2], primes[j])
hodge = data['hodge']
prop2 = [
('Degree', '\(%s\)' % data['degree']),
('Weight', '\(%s\)' % data['weight']),
('Conductor', '\(%s\)' % data['cond']),
]
info.update({
'A': A,
'B': B,
't': t,
'degree': data['degree'],
'weight': data['weight'],
'sign': data['sign'],
'sig': data['sig'],
'hodge': hodge,
'cond': data['cond'],
'req': data['req'],
'locinfo': locinfo
})
friends = []
# friends = [('Galois group', "/GaloisGroup/%dT%d" % (gn, gt))]
# if unramfriend != '':
# friends.append(('Unramified subfield', unramfriend))
# if rffriend != '':
# friends.append(('Discriminant root field', rffriend))
bread = get_bread([(label, ' ')])
return render_template("hgm-show-motive.html", credit=HGM_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends)
def render_hgm_family_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
C = base.getDBConnection()
data = C.hgm.families.find_one({'label': label})
if data is None:
bread = get_bread([("Search error", url_for('.search'))])
info['err'] = "Family of hypergeometric motives " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
title = 'Hypergeometric Motive Family:' + label
A = data['A']
B = data['B']
hodge = data['hodge']
prop2 = [
('Degree', '\(%s\)' % data['degree']),
('Weight', '\(%s\)' % data['weight'])
]
info.update({
'A': A,
'B': B,
'degree': data['degree'],
'weight': data['weight'],
'hodge': hodge,
'gal2': data['gal2'],
'gal3': data['gal3'],
'gal5': data['gal5'],
'gal7': data['gal7'],
})
friends = []
# friends = [('Galois group', "/GaloisGroup/%dT%d" % (gn, gt))]
# if unramfriend != '':
# friends.append(('Unramified subfield', unramfriend))
# if rffriend != '':
# friends.append(('Discriminant root field', rffriend))
info.update({"plotcircle": url_for(".hgm_family_circle_image", AB = "A"+".".join(map(str,A))+"_B"+".".join(map(str,B)))})
info.update({"plotlinear": url_for(".hgm_family_linear_image", AB = "A"+".".join(map(str,A))+"_B"+".".join(map(str,B)))})
info.update({"plotconstant": url_for(".hgm_family_constant_image", AB = "A"+".".join(map(str,A))+"_B"+".".join(map(str,B)))})
bread = get_bread([(label, ' ')])
return render_template("hgm-show-family.html", credit=HGM_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends)
def render_field_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
data = db.lf_fields.lookup(label)
if data is None:
if re.match(r'^\d+\.\d+\.\d+\.\d+$', label):
flash_error("Field <span style='color:black'>%s</span> was not found in the database.", label)
else:
flash_error("<span style='color:black'>%s</span> is not a valid label for a local number field.", label)
return redirect(url_for(".index"))
title = 'Local Number Field ' + prettyname(data)
polynomial = coeff_to_poly(data['coeffs'])
p = data['p']
Qp = r'\Q_{%d}' % p
e = data['e']
f = data['f']
cc = data['c']
gt = data['gal']
gn = data['n']
the_gal = WebGaloisGroup.from_nt(gn,gt)
isgal = ' Galois' if the_gal.order() == gn else ' not Galois'
abelian = ' and abelian' if the_gal.is_abelian() else ''
galphrase = 'This field is'+isgal+abelian+' over $\Q_{%d}$.'%p
autstring = r'\Gal' if the_gal.order() == gn else r'\Aut'
prop2 = [
('Label', label),
('Base', '\(%s\)' % Qp),
('Degree', '\(%s\)' % data['n']),
('e', '\(%s\)' % e),
('f', '\(%s\)' % f),
('c', '\(%s\)' % cc),
('Galois group', group_pretty_and_nTj(gn, gt)),
]
# Look up the unram poly so we can link to it
unramlabel = db.lf_fields.lucky({'p': p, 'n': f, 'c': 0}, projection=0)
if unramlabel is None:
logger.fatal("Cannot find unramified field!")
unramfriend = ''
else:
unramfriend = "/LocalNumberField/%s" % unramlabel
unramdata = db.lf_fields.lookup(unramlabel)
Px = PolynomialRing(QQ, 'x')
Pxt=PolynomialRing(Px,'t')
Pt = PolynomialRing(QQ, 't')
Ptx = PolynomialRing(Pt, 'x')
if data['f'] == 1:
unramp = r'$%s$' % Qp
# Eliminate t from the eisenstein polynomial
eisenp = Pxt(str(data['eisen']).replace('y','x'))
eisenp = Pt(str(data['unram'])).resultant(eisenp)
eisenp = web_latex(eisenp)
else:
unramp = data['unram'].replace('t','x')
unramp = web_latex(Px(str(unramp)))
unramp = prettyname(unramdata)+' $\\cong '+Qp+'(t)$ where $t$ is a root of '+unramp
eisenp = Ptx(str(data['eisen']).replace('y','x'))
eisenp = '$'+web_latex(eisenp, False)+'\\in'+Qp+'(t)[x]$'
rflabel = db.lf_fields.lucky({'p': p, 'n': {'$in': [1, 2]}, 'rf': data['rf']}, projection=0)
if rflabel is None:
logger.fatal("Cannot find discriminant root field!")
rffriend = ''
else:
rffriend = "/LocalNumberField/%s" % rflabel
gsm = data['gsm']
if gsm == [0]:
gsm = 'Not computed'
elif gsm == [-1]:
gsm = 'Does not exist'
else:
gsm = web_latex(coeff_to_poly(gsm))
info.update({
'polynomial': web_latex(polynomial),
'n': data['n'],
'p': p,
'c': data['c'],
'e': data['e'],
'f': data['f'],
't': data['t'],
'u': data['u'],
'rf': lf_display_knowl( rflabel, name=printquad(data['rf'], p)),
'base': lf_display_knowl(str(p)+'.1.0.1', name='$%s$'%Qp),
'hw': data['hw'],
'slopes': show_slopes(data['slopes']),
'gal': group_pretty_and_nTj(gn, gt, True),
'gt': gt,
'inertia': group_display_inertia(data['inertia']),
'unram': unramp,
'eisen': eisenp,
'gms': data['gms'],
'gsm': gsm,
'galphrase': galphrase,
'autstring': autstring,
'subfields': format_subfields(data['subfields'],p),
'aut': data['aut'],
})
friends = [('Galois group', "/GaloisGroup/%dT%d" % (gn, gt))]
if unramfriend != '':
friends.append(('Unramified subfield', unramfriend))
if rffriend != '':
friends.append(('Discriminant root field', rffriend))
bread = get_bread([(label, ' ')])
return render_template("lf-show-field.html", credit=LF_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends, learnmore=learnmore_list())
def hgm_search(**args):
info = to_dict(args)
bread = get_bread([("Search results", url_for('.search'))])
C = base.getDBConnection()
query = {}
if 'jump_to' in info:
return render_hgm_webpage({'label': info['jump_to']})
# t, generic, irreducible
# 'A', 'B', 'hodge'
for param in ['t', 'A', 'B']:
if (param == 't' and PAIR_RE.match(info['t'])) or (param == 'A' and IF_RE.match(info[param])) or (param == 'B' and IF_RE.match(info[param])):
query[param] = parse_list(info[param])
else:
print "Bad input"
for param in ['degree','weight','sign']:
if info.get(param):
info[param] = clean_input(info[param])
if param == 'gal':
try:
gcs = complete_group_codes(info[param])
if len(gcs) == 1:
tmp = ['gal', list(gcs[0])]
if len(gcs) > 1:
tmp = [{'gal': list(x)} for x in gcs]
tmp = ['$or', tmp]
except NameError as code:
info['err'] = 'Error parsing input for A: unknown group label %s. It needs to be a <a title = "Galois group labels" knowl="nf.galois_group.name">group label</a>, such as C5 or 5T1, or comma separated list of labels.' % code
return search_input_error(info, bread)
else:
ran = info[param]
ran = ran.replace('..', '-')
if LIST_RE.match(ran):
tmp = parse_range2(ran, param)
else:
names = {'weight': 'weight', 'degree': 'degree', 'sign': 'sign'}
info['err'] = 'Error parsing input for the %s. It needs to be an integer (such as 5), a range of integers (such as 2-10 or 2..10), or a comma-separated list of these (such as 2,3,8 or 3-5, 7, 8-11).' % names[param]
return search_input_error(info, bread)
# work around syntax for $or
# we have to foil out multiple or conditions
if tmp[0] == '$or' and '$or' in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query['$or']]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
query[tmp[0]] = tmp[1]
count_default = 20
if info.get('count'):
try:
count = int(info['count'])
except:
count = count_default
else:
count = count_default
info['count'] = count
start_default = 0
if info.get('start'):
try:
start = int(info['start'])
if(start < 0):
start += (1 - (start + 1) / count) * count
except:
start = start_default
else:
start = start_default
if info.get('paging'):
try:
paging = int(info['paging'])
if paging == 0:
start = 0
except:
pass
# logger.debug(query)
res = C.hgm.motives.find(query).sort([('degree', pymongo.ASCENDING),
('label', pymongo.ASCENDING)])
nres = res.count()
res = res.skip(start).limit(count)
if(start >= nres):
start -= (1 + (start - nres) / count) * count
if(start < 0):
start = 0
info['motives'] = res
info['number'] = nres
info['start'] = start
if nres == 1:
info['report'] = 'unique match'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
return render_template("hgm-search.html", info=info, title="Hypergeometric Motive over $\Q$ Search Result", bread=bread, credit=HGM_credit)
def render_field_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
data = db.lf_fields.lookup(label)
if data is None:
if re.match(r'^\d+\.\d+\.\d+\.\d+$', label):
flash_error("Field %s was not found in the database.", label)
else:
flash_error("%s is not a valid label for a $p$-adic field.",
label)
return redirect(url_for(".index"))
title = '$p$-adic field ' + prettyname(data)
titletag = 'p-adic field ' + prettyname(data)
polynomial = coeff_to_poly(data['coeffs'])
p = data['p']
Qp = r'\Q_{%d}' % p
e = data['e']
f = data['f']
cc = data['c']
gt = int(data['galois_label'].split('T')[1])
gn = data['n']
the_gal = WebGaloisGroup.from_nt(gn, gt)
isgal = ' Galois' if the_gal.order() == gn else ' not Galois'
abelian = ' and abelian' if the_gal.is_abelian() else ''
galphrase = 'This field is' + isgal + abelian + r' over $\Q_{%d}.$' % p
autstring = r'\Gal' if the_gal.order() == gn else r'\Aut'
prop2 = [
('Label', label),
('Base', r'\(%s\)' % Qp),
('Degree', r'\(%s\)' % data['n']),
('e', r'\(%s\)' % e),
('f', r'\(%s\)' % f),
('c', r'\(%s\)' % cc),
('Galois group', group_pretty_and_nTj(gn, gt)),
]
# Look up the unram poly so we can link to it
unramlabel = db.lf_fields.lucky({'p': p, 'n': f, 'c': 0}, projection=0)
if unramlabel is None:
logger.fatal("Cannot find unramified field!")
unramfriend = ''
else:
unramfriend = "/LocalNumberField/%s" % unramlabel
unramdata = db.lf_fields.lookup(unramlabel)
Px = PolynomialRing(QQ, 'x')
Pt = PolynomialRing(QQ, 't')
Ptx = PolynomialRing(Pt, 'x')
if data['f'] == 1:
unramp = r'$%s$' % Qp
eisenp = Ptx(str(data['eisen']).replace('y', 'x'))
eisenp = web_latex(eisenp)
else:
unramp = data['unram'].replace('t', 'x')
unramp = web_latex(Px(str(unramp)))
unramp = prettyname(
unramdata
) + ' $\\cong ' + Qp + '(t)$ where $t$ is a root of ' + unramp
eisenp = Ptx(str(data['eisen']).replace('y', 'x'))
eisenp = '$' + web_latex(eisenp, False) + '\\in' + Qp + '(t)[x]$'
rflabel = db.lf_fields.lucky(
{
'p': p,
'n': {
'$in': [1, 2]
},
'rf': data['rf']
}, projection=0)
if rflabel is None:
logger.fatal("Cannot find discriminant root field!")
rffriend = ''
else:
rffriend = "/LocalNumberField/%s" % rflabel
gsm = data['gsm']
if gsm == [0]:
gsm = 'Not computed'
elif gsm == [-1]:
gsm = 'Does not exist'
else:
gsm = lf_formatfield(','.join([str(b) for b in gsm]))
info.update({
'polynomial':
web_latex(polynomial),
'n':
data['n'],
'p':
p,
'c':
data['c'],
'e':
data['e'],
'f':
data['f'],
't':
data['t'],
'u':
data['u'],
'rf':
lf_display_knowl(rflabel, name=printquad(data['rf'], p)),
'base':
lf_display_knowl(str(p) + '.1.0.1', name='$%s$' % Qp),
'hw':
data['hw'],
'slopes':
show_slopes(data['slopes']),
'gal':
group_pretty_and_nTj(gn, gt, True),
'gt':
gt,
'inertia':
group_display_inertia(data['inertia']),
'unram':
unramp,
'eisen':
eisenp,
'gms':
data['gms'],
'gsm':
gsm,
'galphrase':
galphrase,
'autstring':
autstring,
'subfields':
format_subfields(data['subfields'], p),
'aut':
data['aut'],
})
friends = [('Galois group', "/GaloisGroup/%dT%d" % (gn, gt))]
if unramfriend != '':
friends.append(('Unramified subfield', unramfriend))
if rffriend != '':
friends.append(('Discriminant root field', rffriend))
bread = get_bread([(label, ' ')])
return render_template(
"lf-show-field.html",
credit=LF_credit,
title=title,
titletag=titletag,
bread=bread,
info=info,
properties=prop2,
friends=friends,
learnmore=learnmore_list(),
KNOWL_ID="lf.%s" % label,
)
def elliptic_curve_search(**args):
info = to_dict(args)
query = {}
bread = [('Elliptic Curves', url_for("ecnf.index")),
('$\Q$', url_for(".rational_elliptic_curves")),
('Search Results', '.')]
if 'SearchAgain' in args:
return rational_elliptic_curves()
if 'jump' in args:
label = info.get('label', '').replace(" ", "")
m = match_lmfdb_label(label)
if m:
try:
return by_ec_label(label)
except ValueError:
return elliptic_curve_jump_error(label, info, wellformed_label=True)
elif label.startswith("Cremona:"):
label = label[8:]
m = match_cremona_label(label)
if m:
try:
return by_ec_label(label)
except ValueError:
return elliptic_curve_jump_error(label, info, wellformed_label=True)
elif match_cremona_label(label):
return elliptic_curve_jump_error(label, info, cremona_label=True)
elif label:
# Try to parse a string like [1,0,3,2,4] as valid
# Weistrass coefficients:
lab = re.sub(r'\s','',label)
lab = re.sub(r'^\[','',lab)
lab = re.sub(r']$','',lab)
try:
labvec = lab.split(',')
labvec = [QQ(str(z)) for z in labvec] # Rationals allowed
E = EllipticCurve(labvec)
# Now we do have a valid curve over Q, but it might
# not be in the database.
ainvs = [str(c) for c in E.minimal_model().ainvs()]
data = db_ec().find_one({'ainvs': ainvs})
if data is None:
info['conductor'] = E.conductor()
return elliptic_curve_jump_error(label, info, missing_curve=True)
return by_ec_label(data['lmfdb_label'])
except (TypeError, ValueError, ArithmeticError):
return elliptic_curve_jump_error(label, info)
else:
query['label'] = ''
if info.get('jinv'):
j = clean_input(info['jinv'])
j = j.replace('+', '')
if not QQ_RE.match(j):
info['err'] = 'Error parsing input for the j-invariant. It needs to be a rational number.'
return search_input_error(info, bread)
query['jinv'] = str(QQ(j)) # to simplify e.g. 1728/1
for field in ['conductor', 'torsion', 'rank', 'sha']:
if info.get(field):
info[field] = clean_input(info[field])
ran = info[field]
ran = ran.replace('..', '-').replace(' ', '')
if not LIST_RE.match(ran):
names = {'conductor': 'conductor', 'torsion': 'torsion order', 'rank':
'rank', 'sha': 'analytic order of Ш'}
info['err'] = 'Error parsing input for the %s. It needs to be an integer (such as 25), a range of integers (such as 2-10 or 2..10), or a comma-separated list of these (such as 4,9,16 or 4-25, 81-121).' % names[field]
return search_input_error(info, bread)
# Past input check
tmp = parse_range2(ran, field)
# work around syntax for $or
# we have to foil out multiple or conditions
if tmp[0] == '$or' and '$or' in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query['$or']]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
query[tmp[0]] = tmp[1]
if 'optimal' in info and info['optimal'] == 'on':
# fails on 990h3
query['number'] = 1
if 'torsion_structure' in info and info['torsion_structure']:
res = parse_torsion_structure(info['torsion_structure'],2)
if 'Error' in res:
info['err'] = res
return search_input_error(info, bread)
#update info for repeat searches
info['torsion_structure'] = str(res).replace(' ','')
query['torsion_structure'] = [str(r) for r in res]
if info.get('surj_primes'):
info['surj_primes'] = clean_input(info['surj_primes'])
format_ok = LIST_POSINT_RE.match(info['surj_primes'])
if format_ok:
surj_primes = [int(p) for p in info['surj_primes'].split(',')]
format_ok = all([ZZ(p).is_prime(proof=False) for p in surj_primes])
if format_ok:
query['non-surjective_primes'] = {"$nin": surj_primes}
else:
info['err'] = 'Error parsing input for surjective primes. It needs to be a prime (such as 5), or a comma-separated list of primes (such as 2,3,11).'
return search_input_error(info, bread)
if info.get('nonsurj_primes'):
info['nonsurj_primes'] = clean_input(info['nonsurj_primes'])
format_ok = LIST_POSINT_RE.match(info['nonsurj_primes'])
if format_ok:
nonsurj_primes = [int(p) for p in info['nonsurj_primes'].split(',')]
format_ok = all([ZZ(p).is_prime(proof=False) for p in nonsurj_primes])
if format_ok:
if info['surj_quantifier'] == 'exactly':
nonsurj_primes.sort()
query['non-surjective_primes'] = nonsurj_primes
else:
if 'non-surjective_primes' in query:
query['non-surjective_primes'] = { "$nin": surj_primes, "$all": nonsurj_primes }
else:
query['non-surjective_primes'] = { "$all": nonsurj_primes }
else:
info['err'] = 'Error parsing input for nonsurjective primes. It needs to be a prime (such as 5), or a comma-separated list of primes (such as 2,3,11).'
return search_input_error(info, bread)
if 'download' in info and info['download'] != '0':
res = db_ec().find(query).sort([ ('conductor', ASCENDING), ('iso_nlabel', ASCENDING), ('lmfdb_number', ASCENDING) ])
return download_search(info, res)
count_default = 100
if info.get('count'):
try:
count = int(info['count'])
except:
count = count_default
else:
count = count_default
info['count'] = count
start_default = 0
if info.get('start'):
try:
start = int(info['start'])
if(start < 0):
start += (1 - (start + 1) / count) * count
except:
start = start_default
else:
start = start_default
cursor = db_ec().find(query)
nres = cursor.count()
if(start >= nres):
start -= (1 + (start - nres) / count) * count
if(start < 0):
start = 0
res = cursor.sort([('conductor', ASCENDING), ('iso_nlabel', ASCENDING), ('lmfdb_number', ASCENDING)
]).skip(start).limit(count)
info['curves'] = res
info['format_ainvs'] = format_ainvs
info['curve_url'] = lambda dbc: url_for(".by_triple_label", conductor=dbc['conductor'], iso_label=split_lmfdb_label(dbc['lmfdb_iso'])[1], number=dbc['lmfdb_number'])
info['iso_url'] = lambda dbc: url_for(".by_double_iso_label", conductor=dbc['conductor'], iso_label=split_lmfdb_label(dbc['lmfdb_iso'])[1])
info['number'] = nres
info['start'] = start
info['count'] = count
info['more'] = int(start + count < nres)
if nres == 1:
info['report'] = 'unique match'
elif nres == 2:
info['report'] = 'displaying both matches'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
credit = 'John Cremona'
if 'non-surjective_primes' in query:
credit += 'and Andrew Sutherland'
t = 'Elliptic Curves search results'
return render_template("search_results.html", info=info, credit=credit, bread=bread, title=t)
def parse_artin_label(label):
label = clean_input(label)
if LABEL_RE.match(label):
return label
else:
raise ValueError
def parse_artin_label(label):
label = clean_input(label)
if LABEL_RE.match(label):
return label
else:
raise ValueError("Error parsing input %s. It is not in a valid form for an Artin representation label, such as 9.2e12_587e3.10t32.1c1"% label)
def elliptic_curve_search(**args):
info = to_dict(args)
query = {}
bread = [("Elliptic Curves", url_for("rational_elliptic_curves")), ("Search Results", ".")]
if "jump" in args:
label = info.get("label", "").replace(" ", "")
m = lmfdb_label_regex.match(label)
if m:
try:
return by_ec_label(label)
except ValueError:
return elliptic_curve_jump_error(label, info, wellformed_label=True)
elif label.startswith("Cremona:"):
label = label[8:]
m = cremona_label_regex.match(label)
if m:
try:
return by_ec_label(label)
except ValueError:
return elliptic_curve_jump_error(label, info, wellformed_label=True)
elif cremona_label_regex.match(label):
return elliptic_curve_jump_error(label, info, cremona_label=True)
elif label:
# Try to parse a string like [1,0,3,2,4]
lab = re.sub(r"\s", "", label)
lab = re.sub(r"^\[", "", lab)
lab = re.sub(r"]$", "", lab)
try:
labvec = lab.split(",")
labvec = [QQ(str(z)) for z in labvec] # Rationals allowed
E = EllipticCurve(labvec)
ainvs = [str(c) for c in E.minimal_model().ainvs()]
C = lmfdb.base.getDBConnection()
data = C.elliptic_curves.curves.find_one({"ainvs": ainvs})
if data is None:
return elliptic_curve_jump_error(label, info)
return by_ec_label(data["lmfdb_label"])
except (ValueError, ArithmeticError):
return elliptic_curve_jump_error(label, info)
else:
query["label"] = ""
if info.get("jinv"):
j = clean_input(info["jinv"])
j = j.replace("+", "")
if not QQ_RE.match(j):
info["err"] = "Error parsing input for the j-invariant. It needs to be a rational number."
return search_input_error(info, bread)
query["jinv"] = j
for field in ["conductor", "torsion", "rank", "sha_an"]:
if info.get(field):
info[field] = clean_input(info[field])
ran = info[field]
ran = ran.replace("..", "-").replace(" ", "")
if not LIST_RE.match(ran):
names = {
"conductor": "conductor",
"torsion": "torsion order",
"rank": "rank",
"sha_an": "analytic order of Ш",
}
info["err"] = (
"Error parsing input for the %s. It needs to be an integer (such as 5), a range of integers (such as 2-10 or 2..10), or a comma-separated list of these (such as 2,3,8 or 3-5, 7, 8-11)."
% names[field]
)
return search_input_error(info, bread)
# Past input check
tmp = parse_range2(ran, field)
# work around syntax for $or
# we have to foil out multiple or conditions
if tmp[0] == "$or" and "$or" in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query["$or"]]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
if field == "sha_an": # database sha_an values are not all exact!
query[tmp[0]] = {"$gt": tmp[1] - 0.1, "$lt": tmp[1] + 0.1}
print query
else:
query[tmp[0]] = tmp[1]
if "optimal" in info and info["optimal"] == "on":
# fails on 990h3
query["number"] = 1
if "torsion_structure" in info and info["torsion_structure"]:
info["torsion_structure"] = clean_input(info["torsion_structure"])
if not TORS_RE.match(info["torsion_structure"]):
info[
"err"
] = "Error parsing input for the torsion structure. It needs to be one or more integers in square brackets, such as [6], [2,2], or [2,4]. Moreover, each integer should be bigger than 1, and each divides the next."
return search_input_error(info, bread)
query["torsion_structure"] = [str(a) for a in parse_list(info["torsion_structure"])]
info["query"] = query
count_default = 100
if info.get("count"):
try:
count = int(info["count"])
except:
count = count_default
else:
count = count_default
info["count"] = count
start_default = 0
if info.get("start"):
try:
start = int(info["start"])
if start < 0:
start += (1 - (start + 1) / count) * count
except:
start = start_default
else:
start = start_default
print query
cursor = lmfdb.base.getDBConnection().elliptic_curves.curves.find(query)
nres = cursor.count()
if start >= nres:
start -= (1 + (start - nres) / count) * count
if start < 0:
start = 0
res = (
cursor.sort([("conductor", ASCENDING), ("lmfdb_iso", ASCENDING), ("lmfdb_number", ASCENDING)])
.skip(start)
.limit(count)
)
info["curves"] = res
info["format_ainvs"] = format_ainvs
info["number"] = nres
info["start"] = start
if nres == 1:
info["report"] = "unique match"
else:
if nres > count or start != 0:
info["report"] = "displaying matches %s-%s of %s" % (start + 1, min(nres, start + count), nres)
else:
info["report"] = "displaying all %s matches" % nres
credit = "John Cremona"
t = "Elliptic Curves"
return render_template("elliptic_curve/elliptic_curve_search.html", info=info, credit=credit, bread=bread, title=t)
def elliptic_curve_search(**args):
info = to_dict(args["data"])
if "jump" in info:
label = info.get("label", "").replace(" ", "")
# This label should be a full isogeny class label or a full
# curve label (including the field_label component)
try:
nf, cond_label, iso_label, number = split_full_label(label)
except IndexError:
if not "query" in info:
info["query"] = {}
bread = [("Elliptic Curves", url_for(".index"))]
info["err"] = "No elliptic curve in the database has label %s." % label
return search_input_error(info, bread)
return show_ecnf(nf, cond_label, iso_label, number)
query = {}
bread = [("Elliptic Curves", url_for(".index")), ("Search Results", ".")]
if "conductor_norm" in info:
Nnorm = clean_input(info["conductor_norm"])
Nnorm = Nnorm.replace("..", "-").replace(" ", "")
tmp = parse_range2(Nnorm, "conductor_norm")
if tmp[0] == "$or" and "$or" in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query["$or"]]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
query[tmp[0]] = tmp[1]
if "conductor_label" in info:
query["conductor_label"] = info["conductor_label"]
if "jinv" in info:
query["jinv"] = info["jinv"]
if info.get("torsion"):
ran = info["torsion"] = clean_input(info["torsion"])
ran = ran.replace("..", "-").replace(" ", "")
if not LIST_RE.match(ran):
info[
"err"
] = "Error parsing input for the torsion order. It needs to be an integer (such as 5), a range of integers (such as 2-10 or 2..10), or a comma-separated list of these (such as 4,9,16 or 4-25, 81-121)."
return search_input_error(info, bread)
# Past input check
tmp = parse_range2(ran, "torsion_order")
# work around syntax for $or
# we have to foil out multiple or conditions
if tmp[0] == "$or" and "$or" in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query["$or"]]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
query[tmp[0]] = tmp[1]
if "torsion_structure" in info and info["torsion_structure"]:
res = parse_torsion_structure(info["torsion_structure"], 2)
if "Error" in res:
info["err"] = res
return search_input_error(info, bread)
# update info for repeat searches
info["torsion_structure"] = str(res).replace(" ", "")
query["torsion_structure"] = [int(r) for r in res]
if "include_isogenous" in info and info["include_isogenous"] == "off":
query["number"] = 1
if "include_base_change" in info and info["include_base_change"] == "off":
query["base_change"] = []
else:
info["include_base_change"] = "on"
if "field" in info:
query["field_label"] = parse_field_string(info["field"])
info["query"] = query
# process count and start if not default:
count_default = 50
if info.get("count"):
try:
count = int(info["count"])
except:
count = count_default
else:
count = count_default
start_default = 0
if info.get("start"):
try:
start = int(info["start"])
if start < 0:
start += (1 - (start + 1) / count) * count
except:
start = start_default
else:
start = start_default
# make the query and trim results according to start/count:
cursor = db_ecnf().find(query)
nres = cursor.count()
if start >= nres:
start -= (1 + (start - nres) / count) * count
if start < 0:
start = 0
res = (
cursor.sort(
[
("field_label", ASC),
("conductor_norm", ASC),
("conductor_label", ASC),
("iso_label", ASC),
("number", ASC),
]
)
.skip(start)
.limit(count)
)
res = list(res)
for e in res:
e["numb"] = str(e["number"])
e["field_knowl"] = nf_display_knowl(e["field_label"], getDBConnection(), e["field_label"])
info["curves"] = res # [ECNF(e) for e in res]
info["number"] = nres
info["start"] = start
info["count"] = count
info["more"] = int(start + count < nres)
info["field_pretty"] = field_pretty
info["web_ainvs"] = web_ainvs
if nres == 1:
info["report"] = "unique match"
else:
if nres > count or start != 0:
info["report"] = "displaying matches %s-%s of %s" % (start + 1, min(nres, start + count), nres)
else:
info["report"] = "displaying all %s matches" % nres
t = "Elliptic Curve search results"
return render_template("ecnf-search-results.html", info=info, credit=ecnf_credit, bread=bread, title=t)
def elliptic_curve_search(**args):
info = to_dict(args['data'])
if 'jump' in info:
label = info.get('label', '').replace(" ", "")
# This label should be a full isogeny class label or a full
# curve label (including the field_label component)
try:
nf, cond_label, iso_label, number = split_full_label(label)
except IndexError:
if not 'query' in info:
info['query'] = {}
bread = [("Elliptic Curves", url_for(".index"))]
info['err'] = 'No elliptic curve in the database has label %s.' % label
return search_input_error(info, bread)
return show_ecnf(nf, cond_label, iso_label, number)
query = {}
bread = [('Elliptic Curves', url_for(".index")),
('Search Results', '.')]
if 'conductor_norm' in info:
Nnorm = clean_input(info['conductor_norm'])
Nnorm = Nnorm.replace('..', '-').replace(' ', '')
tmp = parse_range2(Nnorm, 'conductor_norm')
if tmp[0] == '$or' and '$or' in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query['$or']]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
query[tmp[0]] = tmp[1]
if 'conductor_label' in info:
query['conductor_label'] = info['conductor_label']
if 'jinv' in info:
query['jinv'] = info['jinv']
if info.get('torsion'):
ran = info['torsion'] = clean_input(info['torsion'])
ran = ran.replace('..', '-').replace(' ', '')
if not LIST_RE.match(ran):
info['err'] = 'Error parsing input for the torsion order. It needs to be an integer (such as 5), a range of integers (such as 2-10 or 2..10), or a comma-separated list of these (such as 4,9,16 or 4-25, 81-121).'
return search_input_error(info, bread)
# Past input check
tmp = parse_range2(ran, 'torsion_order')
# work around syntax for $or
# we have to foil out multiple or conditions
if tmp[0] == '$or' and '$or' in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query['$or']]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
query[tmp[0]] = tmp[1]
if 'torsion_structure' in info and info['torsion_structure']:
info['torsion_structure'] = clean_input(info['torsion_structure'])
if not TORS_RE.match(info['torsion_structure']):
info['err'] = 'Error parsing input for the torsion structure. It needs to be one or more integers in square brackets, such as [6], [2,2], or [2,4]. Moreover, each integer should be bigger than 1, and each divides the next.'
return search_input_error(info, bread)
query['torsion_structure'] = parse_list(info['torsion_structure'])
if 'include_isogenous' in info and info['include_isogenous'] == 'off':
query['number'] = 1
if 'include_base_change' in info and info['include_base_change'] == 'off':
query['base_change'] = []
else:
info['include_base_change'] = "on"
if 'field' in info:
query['field_label'] = parse_field_string(info['field'])
info['query'] = query
# process count and start if not default:
count_default = 50
if info.get('count'):
try:
count = int(info['count'])
except:
count = count_default
else:
count = count_default
start_default = 0
if info.get('start'):
try:
start = int(info['start'])
if(start < 0):
start += (1 - (start + 1) / count) * count
except:
start = start_default
else:
start = start_default
# make the query and trim results according to start/count:
cursor = db_ecnf().find(query)
nres = cursor.count()
if(start >= nres):
start -= (1 + (start - nres) / count) * count
if(start < 0):
start = 0
res = cursor.sort([('field_label', ASC), ('conductor_norm', ASC), ('conductor_label', ASC), ('iso_label', ASC), ('number', ASC)]).skip(start).limit(count)
res = list(res)
for e in res:
e['numb'] = str(e['number'])
e['field_knowl'] = nf_display_knowl(e['field_label'], getDBConnection(), e['field_label'])
info['curves'] = res # [ECNF(e) for e in res]
info['number'] = nres
info['start'] = start
info['count'] = count
info['more'] = int(start + count < nres)
info['field_pretty'] = field_pretty
info['web_ainvs'] = web_ainvs
if nres == 1:
info['report'] = 'unique match'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
t = 'Elliptic Curve search results'
return render_template("ecnf-search-results.html", info=info, credit=ecnf_credit, bread=bread, title=t)
def number_field_search(**args):
info = to_dict(args)
info["learnmore"] = [
("Global number field labels", url_for(".render_labels_page")),
("Galois group labels", url_for(".render_groups_page")),
(Completename, url_for(".render_discriminants_page")),
]
t = "Global Number Field search results"
bread = [("Global Number Fields", url_for(".number_field_render_webpage")), ("Search results", " ")]
# for k in info.keys():
# nf_logger.debug(str(k) + ' ---> ' + str(info[k]))
# nf_logger.debug('******************* '+ str(info['search']))
if "natural" in info:
field_id = info["natural"]
field_id_parsed = parse_field_string(info["natural"])
if FIELD_LABEL_RE.match(field_id_parsed):
field_id_parsed = split_label(field_id_parsed) # allows factored labels 11.11.11e20.1
return render_field_webpage({"label": field_id_parsed, "label_orig": field_id})
query = {}
dlist = []
for field in ["galois_group", "degree", "signature", "discriminant", "class_number", "class_group"]:
if info.get(field):
info[field] = clean_input(info[field])
if field in ["class_group", "signature"]:
# different regex for the two types
if (field == "signature" and PAIR_RE.match(info[field])) or (
field == "class_group" and IF_RE.match(info[field])
):
query[field] = info[field][1:-1]
else:
name = "class group" if field == "class_group" else "signature"
info["err"] = (
"Error parsing input for %s. It needs to be a pair of integers in square brackets, such as [2,3] or [3,3]"
% name
)
return search_input_error(info, bread)
else:
if field == "galois_group":
try:
gcs = complete_group_codes(info[field])
if len(gcs) == 1:
query["galois"] = make_galois_pair(gcs[0][0], gcs[0][1])
# list(gcs[0])
if len(gcs) > 1:
query["galois"] = {"$in": [make_galois_pair(x[0], x[1]) for x in gcs]}
except NameError as code:
info["err"] = (
'Error parsing input for Galois group: unknown group label %s. It needs to be a <a title = "Galois group labels" knowl="nf.galois_group.name">group label</a>, such as C5 or 5T1, or comma separated list of labels.'
% code
)
return search_input_error(info, bread)
else: # not signature, class group, or galois group
ran = info[field]
ran = ran.replace("..", "-")
if LIST_RE.match(ran):
if field == "discriminant":
# dlist will contain the disc conditions
# as sage ints
dlist = parse_discs(ran)
# now convert to a query
tmp = list_to_query(dlist)
# Two cases, could be a list of sign/inequalties
# or an '$or'
if len(tmp) == 1:
tmp = tmp[0]
else:
query[tmp[0][0]] = tmp[0][1]
tmp = tmp[1]
else:
tmp = parse_range2(ran, field)
# work around syntax for $or
# we have to foil out multiple or conditions
if tmp[0] == "$or" and "$or" in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query["$or"]]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
query[tmp[0]] = tmp[1]
else:
name = re.sub("_", " ", field)
info["err"] = (
"Error parsing input for %s. It needs to be an integer (such as 5), a range of integers (such as 2-100 or 2..100), or a comma-separated list of these (such as 2,3,8 or 3-5, 7, 8-100)."
% name
)
return search_input_error(info, bread)
if info.get("ur_primes"):
# now we want a list of strings, no spaces, which might be big ints
info["ur_primes"] = clean_input(info["ur_primes"])
if LIST_SIMPLE_RE.match(info["ur_primes"]):
ur_primes = info["ur_primes"].split(",")
# Assuming this will be the only nor in the query
query["$nor"] = [{"ramps": x} for x in ur_primes]
else:
info[
"err"
] = "Error parsing input for unramified primes. It needs to be an integer (such as 5), or a comma-separated list of integers (such as 2,3,11)."
return search_input_error(info, bread)
if info.get("ram_primes"):
# now we want a list of strings, no spaces, which might be big ints
info["ram_primes"] = clean_input(info["ram_primes"])
if LIST_SIMPLE_RE.match(info["ram_primes"]):
ram_primes = info["ram_primes"].split(",")
if str(info["ram_quantifier"]) == "some":
query["ramps"] = {"$all": ram_primes}
else:
query["ramps"] = ram_primes
else:
info[
"err"
] = "Error parsing input for ramified primes. It needs to be an integer (such as 5), or a comma-separated list of integers (such as 2,3,11)."
return search_input_error(info, bread)
count_default = 20
if info.get("count"):
try:
count = int(info["count"])
except:
count = count_default
info["count"] = count
else:
info["count"] = count_default
count = count_default
info["count"] = int(info["count"])
start_default = 0
if info.get("start"):
try:
start = int(info["start"])
if start < 0:
start += (1 - (start + 1) / count) * count
except:
start = start_default
else:
start = start_default
if info.get("paging"):
try:
paging = int(info["paging"])
if paging == 0:
start = 0
except:
pass
C = base.getDBConnection()
# nf_logger.debug(query)
info["query"] = dict(query)
if "lucky" in args:
one = C.numberfields.fields.find_one(query)
if one:
label = one["label"]
return render_field_webpage({"label": label})
fields = C.numberfields.fields
res = fields.find(query).sort([("degree", ASC), ("disc_abs_key", ASC), ("disc_sign", ASC), ("label", ASC)])
if "download" in info and info["download"] != "0":
return download_search(info, res)
nres = res.count()
res = res.skip(start).limit(count)
if start >= nres:
start -= (1 + (start - nres) / count) * count
if start < 0:
start = 0
info["fields"] = res
info["number"] = nres
info["start"] = start
if nres == 1:
info["report"] = "unique match"
else:
if nres > count or start != 0:
info["report"] = "displaying matches %s-%s of %s" % (start + 1, min(nres, start + count), nres)
else:
info["report"] = "displaying all %s matches" % nres
info["wnf"] = WebNumberField.from_data
return render_template("number_field_search.html", info=info, title=t, bread=bread)
def render_field_webpage(args):
data = None
C = base.getDBConnection()
info = {}
bread = [("Global Number Fields", url_for(".number_field_render_webpage"))]
if "label" in args:
label = clean_input(args["label"])
nf = WebNumberField(label)
data = {}
if nf.is_null():
bread.append(("Search results", " "))
label2 = re.sub(r"[<>]", "", args["label"])
if "You need to enter a field" in label2:
info["err"] = label2
else:
info["err"] = "No such field: %s in the database" % label2
info["label"] = args["label_orig"] if "label_orig" in args else args["label"]
return search_input_error(info, bread)
info["wnf"] = nf
data["degree"] = nf.degree()
data["class_number"] = nf.class_number()
t = nf.galois_t()
n = nf.degree()
data["is_galois"] = nf.is_galois()
data["is_abelian"] = nf.is_abelian()
if nf.is_abelian():
conductor = nf.conductor()
data["conductor"] = conductor
dirichlet_chars = nf.dirichlet_group()
if len(dirichlet_chars) > 0:
data["dirichlet_group"] = [
'<a href = "%s">$\chi_{%s}(%s,·)$</a>'
% (url_character(type="Dirichlet", modulus=data["conductor"], number=j), data["conductor"], j)
for j in dirichlet_chars
]
data["dirichlet_group"] = r"$\lbrace$" + ", ".join(data["dirichlet_group"]) + r"$\rbrace$"
if data["conductor"].is_prime() or data["conductor"] == 1:
data["conductor"] = "\(%s\)" % str(data["conductor"])
else:
data["conductor"] = "\(%s=%s\)" % (str(data["conductor"]), latex(data["conductor"].factor()))
data["galois_group"] = group_display_knowl(n, t, C)
data["cclasses"] = cclasses_display_knowl(n, t, C)
data["character_table"] = character_table_display_knowl(n, t, C)
data["class_group"] = nf.class_group()
data["class_group_invs"] = nf.class_group_invariants()
data["signature"] = nf.signature()
data["coefficients"] = nf.coeffs()
D = nf.disc()
ram_primes = D.prime_factors()
data["disc_factor"] = nf.disc_factored_latex()
if D.abs().is_prime() or D == 1:
data["discriminant"] = "\(%s\)" % str(D)
else:
data["discriminant"] = "\(%s=%s\)" % (str(D), data["disc_factor"])
npr = len(ram_primes)
ram_primes = str(ram_primes)[1:-1]
if ram_primes == "":
ram_primes = r"\textrm{None}"
data["frob_data"], data["seeram"] = frobs(nf.K())
data["phrase"] = group_phrase(n, t, C)
zk = pari(nf.K()).nf_subst("a")
zk = list(zk.nf_get_zk())
Ra = PolynomialRing(QQ, "a")
zk = [latex(Ra(x)) for x in zk]
zk = ["$%s$" % x for x in zk]
zk = ", ".join(zk)
grh_label = (
'<small>(<a title="assuming GRH" knowl="nf.assuming_grh">assuming GRH</a>)</small>' if nf.used_grh() else ""
)
# Short version for properties
grh_lab = nf.short_grh_string()
pretty_label = field_pretty(label)
if label != pretty_label:
pretty_label = "%s: %s" % (label, pretty_label)
info.update(data)
info.update(
{
"label": pretty_label,
"label_raw": label,
"polynomial": web_latex_split_on_pm(nf.K().defining_polynomial()),
"ram_primes": ram_primes,
"integral_basis": zk,
"regulator": web_latex(nf.regulator()),
"unit_rank": nf.unit_rank(),
"root_of_unity": web_latex(nf.K().primitive_root_of_unity()),
"fund_units": nf.units(),
"grh_label": grh_label,
}
)
bread.append(("%s" % info["label_raw"], " "))
info["downloads_visible"] = True
info["downloads"] = [("worksheet", "/")]
info["friends"] = []
if nf.can_class_number():
info["friends"].append(("L-function", "/L/NumberField/%s" % label))
info["friends"].append(("Galois group", "/GaloisGroup/%dT%d" % (n, t)))
if "dirichlet_group" in info:
info["friends"].append(
(
"Dirichlet group",
url_for(
"characters.dirichlet_group_table",
modulus=int(conductor),
char_number_list=",".join([str(a) for a in dirichlet_chars]),
poly=info["polynomial"],
),
)
)
info["learnmore"] = [
("Global number field labels", url_for(".render_labels_page")),
(Completename, url_for(".render_discriminants_page")),
]
# With Galois group labels, probably not needed here
# info['learnmore'] = [('Global number field labels',
# url_for(".render_labels_page")), ('Galois group
# labels',url_for(".render_groups_page")),
# (Completename,url_for(".render_discriminants_page"))]
title = "Global Number Field %s" % info["label"]
if npr == 1:
primes = "prime"
else:
primes = "primes"
properties2 = [
("Degree:", "%s" % data["degree"]),
("Signature:", "$%s$" % data["signature"]),
("Discriminant:", "$%s$" % data["disc_factor"]),
("Ramified " + primes + ":", "$%s$" % ram_primes),
("Class number:", "%s %s" % (data["class_number"], grh_lab)),
("Class group:", "%s %s" % (data["class_group_invs"], grh_lab)),
("Galois Group:", group_display_short(data["degree"], t, C)),
]
from lmfdb.math_classes import NumberFieldGaloisGroup
try:
info["tim_number_field"] = NumberFieldGaloisGroup.find_one({"label": label})
v = nf.factor_perm_repn(info["tim_number_field"])
info["mydecomp"] = ["*" if x > 0 else "" for x in v]
except AttributeError:
pass
# del info['_id']
return render_template(
"number_field.html",
properties2=properties2,
credit=NF_credit,
title=title,
bread=bread,
friends=info.pop("friends"),
learnmore=info.pop("learnmore"),
info=info,
)
def local_field_search(**args):
info = to_dict(args)
bread = get_bread([("Search results", url_for('.search'))])
C = base.getDBConnection()
query = {}
if 'jump_to' in info:
return render_field_webpage({'label': info['jump_to']})
for param in ['p', 'n', 'c', 'e', 'gal']:
if info.get(param):
info[param] = clean_input(info[param])
if param == 'gal':
try:
gcs = complete_group_codes(info[param])
if len(gcs) == 1:
tmp = ['gal', list(gcs[0])]
if len(gcs) > 1:
tmp = [{'gal': list(x)} for x in gcs]
tmp = ['$or', tmp]
except NameError as code:
info['err'] = 'Error parsing input for Galois group: unknown group label %s. It needs to be a <a title = "Galois group labels" knowl="nf.galois_group.name">group label</a>, such as C5 or 5T1, or comma separated list of labels.' % code
return search_input_error(info, bread)
else:
ran = info[param]
ran = ran.replace('..', '-')
if LIST_RE.match(ran):
tmp = parse_range2(ran, param)
else:
names = {'p': 'prime p', 'n': 'degree', 'c':
'discriminant exponent c', 'e': 'ramification index e'}
info['err'] = 'Error parsing input for the %s. It needs to be an integer (such as 5), a range of integers (such as 2-10 or 2..10), or a comma-separated list of these (such as 2,3,8 or 3-5, 7, 8-11).' % names[param]
return search_input_error(info, bread)
# work around syntax for $or
# we have to foil out multiple or conditions
if tmp[0] == '$or' and '$or' in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query['$or']]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
query[tmp[0]] = tmp[1]
count_default = 20
if info.get('count'):
try:
count = int(info['count'])
except:
count = count_default
else:
count = count_default
info['count'] = count
start_default = 0
if info.get('start'):
try:
start = int(info['start'])
if(start < 0):
start += (1 - (start + 1) / count) * count
except:
start = start_default
else:
start = start_default
if info.get('paging'):
try:
paging = int(info['paging'])
if paging == 0:
start = 0
except:
pass
# logger.debug(query)
res = C.localfields.fields.find(query).sort([('p', pymongo.ASCENDING), (
'n', pymongo.ASCENDING), ('c', pymongo.ASCENDING), ('label', pymongo.ASCENDING)])
nres = res.count()
res = res.skip(start).limit(count)
if(start >= nres):
start -= (1 + (start - nres) / count) * count
if(start < 0):
start = 0
info['fields'] = res
info['number'] = nres
info['group_display'] = group_display_shortC(C)
info['display_poly'] = format_coeffs
info['slopedisp'] = show_slope_content
info['start'] = start
if nres == 1:
info['report'] = 'unique match'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
return render_template("lf-search.html", info=info, title="Local Number Field Search Result", bread=bread, credit=LF_credit)
def render_field_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
C = base.getDBConnection()
data = C.localfields.fields.find_one({'label': label})
if data is None:
bread = get_bread([("Search error", url_for('.search'))])
info['err'] = "Field " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
title = 'Local Number Field:' + label
polynomial = coeff_to_poly(data['coeffs'])
p = data['p']
e = data['e']
f = data['f']
cc = data['c']
GG = data['gal']
gn = GG[0]
gt = GG[1]
prop2 = [
('Base', '\(\Q_{%s}\)' % p),
('Degree', '\(%s\)' % data['n']),
('e', '\(%s\)' % e),
('f', '\(%s\)' % f),
('c', '\(%s\)' % cc),
('Galois group', group_display_short(gn, gt, C)),
]
Pt = PolynomialRing(QQ, 't')
Pyt = PolynomialRing(Pt, 'y')
eisenp = Pyt(str(data['eisen']))
unramp = Pyt(str(data['unram']))
# Look up the unram poly so we can link to it
unramdata = C.localfields.fields.find_one({'p': p, 'n': f, 'c': 0})
if len(unramdata) > 0:
unramfriend = "/LocalNumberField/%s" % unramdata['label']
else:
logger.fatal("Cannot find unramified field!")
unramfriend = ''
rfdata = C.localfields.fields.find_one({'p': p, 'n': {'$in': [1, 2]}, 'rf': data['rf']})
if rfdata is None:
logger.fatal("Cannot find discriminant root field!")
rffriend = ''
else:
rffriend = "/LocalNumberField/%s" % rfdata['label']
info.update({
'polynomial': web_latex(polynomial),
'n': data['n'],
'p': data['p'],
'c': data['c'],
'e': data['e'],
'f': data['f'],
't': data['t'],
'u': data['u'],
'rf': printquad(data['rf'], p),
'hw': data['hw'],
'slopes': show_slopes(data['slopes']),
'gal': group_display_knowl(gn, gt, C),
'gt': gt,
'inertia': group_display_inertia(data['inertia'], C),
'unram': web_latex(unramp),
'eisen': web_latex(eisenp),
'gms': data['gms'],
'aut': data['aut'],
})
friends = [('Galois group', "/GaloisGroup/%dT%d" % (gn, gt))]
if unramfriend != '':
friends.append(('Unramified subfield', unramfriend))
if rffriend != '':
friends.append(('Discriminant root field', rffriend))
bread = get_bread([(label, ' ')])
return render_template("lf-show-field.html", credit=LF_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends)
def elliptic_curve_search(**args):
info = to_dict(args["data"])
if "jump" in info:
label = info.get("label", "").replace(" ", "")
# This label should be a full isogeny class label or a full
# curve label (including the field_label component)
label_parts = label.split("-", 2)
nf = label_parts[0]
cond_label = label_parts[1]
cur_label = label_parts[2]
return show_ecnf(nf, cond_label, cur_label) ##!!!
query = {}
if "conductor_norm" in info:
Nnorm = clean_input(info["conductor_norm"])
Nnorm = Nnorm.replace("..", "-").replace(" ", "")
tmp = parse_range2(Nnorm, "conductor_norm")
if tmp[0] == "$or" and "$or" in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query["$or"]]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
query[tmp[0]] = tmp[1]
if "conductor_label" in info:
query["conductor_label"] = info["conductor_label"]
if "include_isogenous" in info and info["include_isogenous"] == "off":
query["number"] = 1
if "field" in info:
query["field_label"] = info["field"]
info["query"] = query
# process count and start if not default:
count_default = 50
if info.get("count"):
try:
count = int(info["count"])
except:
count = count_default
else:
count = count_default
start_default = 0
if info.get("start"):
try:
start = int(info["start"])
if start < 0:
start += (1 - (start + 1) / count) * count
except:
start = start_default
else:
start = start_default
# make the query and trim results according to start/count:
cursor = db_ecnf().find(query)
nres = cursor.count()
if start >= nres:
start -= (1 + (start - nres) / count) * count
if start < 0:
start = 0
res = (
cursor.sort(
[
("field_label", ASC),
("conductor_norm", ASC),
("conductor_label", ASC),
("iso_label", ASC),
("number", ASC),
]
)
.skip(start)
.limit(count)
)
bread = [("Elliptic Curves", url_for(".index")), ("Search Results", ".")]
res = list(res)
for e in res:
e["numb"] = str(e["number"])
e["field_knowl"] = nf_display_knowl(e["field_label"], getDBConnection(), e["field_label"])
info["curves"] = res # [ECNF(e) for e in res]
info["number"] = nres
info["start"] = start
info["count"] = count
info["more"] = int(start + count < nres)
info["field_pretty"] = field_pretty
info["web_ainvs"] = web_ainvs
if nres == 1:
info["report"] = "unique match"
else:
if nres > count or start != 0:
info["report"] = "displaying matches %s-%s of %s" % (start + 1, min(nres, start + count), nres)
else:
info["report"] = "displaying all %s matches" % nres
t = "Elliptic Curve search results"
return render_template("ecnf-search-results.html", info=info, credit=ecnf_credit, bread=bread, title=t)
def galois_group_search(**args):
info = to_dict(args)
bread = get_bread([("Search results", url_for('.search'))])
C = base.getDBConnection()
query = {}
if 'jump_to' in info:
return render_group_webpage({'label': info['jump_to']})
for param in ['n', 't']:
if info.get(param):
info[param] = clean_input(info[param])
ran = info[param]
ran = ran.replace('..', '-')
if LIST_RE.match(ran):
tmp = parse_range2(ran, param)
else:
names = {'n': 'degree', 't': 't'}
info[
'err'] = 'Error parsing input for the %s. It needs to be an integer (such as 5), a range of integers (such as 2-10 or 2..10), or a comma-separated list of these (such as 2,3,8 or 3-5, 7, 8-11).' % names[
param]
return search_input_error(info, bread)
# work around syntax for $or
# we have to foil out multiple or conditions
if tmp[0] == '$or' and '$or' in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query['$or']]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
query[tmp[0]] = tmp[1]
for param in ['cyc', 'solv', 'prim', 'parity']:
if info.get(param):
info[param] = str(info[param])
if info[param] == str(1):
query[param] = 1
elif info[param] == str(-1):
query[param] = -1 if param == 'parity' else 0
# Determine if we have any composite degrees
info['show_subs'] = True
if info.get('n'):
info[
'show_subs'] = False # now only show subs if a composite n is allowed
nparam = info.get('n')
nparam.replace('..', '-')
nlist = nparam.split(',')
found = False
for nl in nlist:
if '-' in nl:
inx = nl.index('-')
ll, hh = nl[:inx], nl[inx + 1:]
hh = int(hh)
jj = int(ll)
while jj <= hh and not found:
if not (ZZ(jj).is_prime() or ZZ(jj) == 1):
found = True
jj += 1
if found:
break
else:
jj = ZZ(nl)
if not (ZZ(jj).is_prime() or ZZ(jj) == 1):
found = True
break
if found:
info['show_subs'] = True
count_default = 20
if info.get('count'):
try:
count = int(info['count'])
except:
count = count_default
else:
count = count_default
info['count'] = count
start_default = 0
if info.get('start'):
try:
start = int(info['start'])
if (start < 0):
start += (1 - (start + 1) / count) * count
except:
start = start_default
else:
start = start_default
if info.get('paging'):
try:
paging = int(info['paging'])
if paging == 0:
start = 0
except:
pass
res = C.transitivegroups.groups.find(query).sort([('n', pymongo.ASCENDING),
('t', pymongo.ASCENDING)
])
nres = res.count()
res = res.skip(start).limit(count)
if (start >= nres):
start -= (1 + (start - nres) / count) * count
if (start < 0):
start = 0
info['groups'] = res
info['group_display'] = group_display_shortC(C)
info['report'] = "found %s groups" % nres
info['yesno'] = yesno
info['wgg'] = WebGaloisGroup.from_data
info['start'] = start
info['number'] = nres
if nres == 1:
info['report'] = 'unique match'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (
start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
return render_template("gg-search.html",
info=info,
title="Galois Group Search Result",
bread=bread,
credit=GG_credit)
def render_hecke_algebras_webpage(**args):
data = None
if 'label' in args:
lab = clean_input(args.get('label'))
if lab != args.get('label'):
return redirect(
url_for('.render_hecke_algebras_webpage', label=lab), 301)
data = db.hecke_algebras.lookup(lab)
if data is None:
t = "Hecke algebra search error"
bread = [('HeckeAlgebra', url_for(".hecke_algebras_render_webpage"))]
flash_error(
"%s is not a valid label for a Hecke algebra in the database.",
lab)
return render_template("hecke_algebras-error.html",
title=t,
properties=[],
bread=bread,
learnmore=learnmore_list())
info = {}
info.update(data)
bread = [('HeckeAlgebra', url_for(".hecke_algebras_render_webpage")),
('%s' % data['label'], ' ')]
credit = hecke_algebras_credit
info['level'] = int(data['level'])
info['weight'] = int(data['weight'])
info['num_orbits'] = int(data['num_orbits'])
dim_count = "not available"
proj = [
'orbit_label', 'hecke_op', 'num_hecke_op', 'Zbasis', 'discriminant',
'disc_fac', 'Qbasis', 'Qalg_gen'
]
orb = list(db.hecke_orbits.search({'parent_label': data['label']}, proj))
if orb:
#consistency check
if len(orb) != int(data['num_orbits']):
return search_input_error(info)
dim_count = 0
for v in orb:
ops = sage_eval(v['hecke_op'])
dim = int(matrix(ops[0]).nrows())
dim_count += dim
if dim > 4:
v['hecke_op_display'] = []
elif dim == 1:
v['hecke_op_display'] = [[i + 1, ops[i][0][0]]
for i in range(10)]
else:
v['hecke_op_display'] = [[i + 1, latex(matrix(ops[i]))]
for i in range(5)]
v['download_op'] = [
(lang,
url_for(".render_hecke_algebras_webpage_download",
orbit_label=v['orbit_label'],
lang=lang,
obj='operators')) for lang in ['gp', 'magma', 'sage']
]
if v['Zbasis'] is None:
for key in [
'Zbasis', 'discriminant', 'disc_fac', 'Qbasis',
'Qalg_gen'
]:
del v[key]
else:
v['Zbasis'] = [[int(i) for i in j] for j in v['Zbasis']]
v['disc_fac'] = [[int(i) for i in j] for j in v['disc_fac']]
if dim > 4:
v['gen_display'] = []
elif dim == 1:
v['gen_display'] = [v['Zbasis'][0][0]]
else:
v['gen_display'] = [
latex(matrix(dim, dim, v['Zbasis'][i]))
for i in range(dim)
]
v['inner_twists'] = "not available" # not yet in the database
v['download_gen'] = [
(lang,
url_for(".render_hecke_algebras_webpage_download",
orbit_label=v['orbit_label'],
lang=lang,
obj='gen')) for lang in ['gp', 'magma', 'sage']
]
info['orbits'] = orb
info['dim_alg'] = dim_count
info['l_adic'] = l_range
info['properties'] = [('Label', '%s' % info['label']),
('Level', '%s' % info['level']),
('Weight', '%s' % info['weight'])]
if info['num_orbits'] != 0:
info['friends'] = [('Newforms space ' + info['label'],
url_for("cmf.by_url_space_label",
level=info['level'],
weight=info['weight'],
char_orbit_label='a'))]
else:
info['friends'] = []
t = "Hecke algebra %s" % info['label']
return render_template("hecke_algebras-single.html",
info=info,
credit=credit,
title=t,
bread=bread,
properties=info['properties'],
learnmore=learnmore_list(),
friends=info['friends'],
KNOWL_ID='hecke_algebra.%s' % (info['label']))
def elliptic_curve_search(**args):
info = to_dict(args)
query = {}
bread = [('Elliptic Curves', url_for("rational_elliptic_curves")),
('Search Results', '.')]
if 'jump' in args:
label = info.get('label', '').replace(" ", "")
m = lmfdb_label_regex.match(label)
if m:
try:
return by_ec_label(label)
except ValueError:
return elliptic_curve_jump_error(label, info, wellformed_label=True)
elif label.startswith("Cremona:"):
label = label[8:]
m = cremona_label_regex.match(label)
if m:
try:
return by_ec_label(label)
except ValueError:
return elliptic_curve_jump_error(label, info, wellformed_label=True)
elif cremona_label_regex.match(label):
return elliptic_curve_jump_error(label, info, cremona_label=True)
elif label:
# Try to parse a string like [1,0,3,2,4]
lab = re.sub(r'\s','',label)
lab = re.sub(r'^\[','',lab)
lab = re.sub(r']$','',lab)
try:
labvec = lab.split(',')
labvec = [QQ(str(z)) for z in labvec] # Rationals allowed
E = EllipticCurve(labvec)
ainvs = [str(c) for c in E.minimal_model().ainvs()]
C = lmfdb.base.getDBConnection()
data = C.elliptic_curves.curves.find_one({'ainvs': ainvs})
if data is None:
return elliptic_curve_jump_error(label, info)
return by_ec_label(data['lmfdb_label'])
except (ValueError, ArithmeticError):
return elliptic_curve_jump_error(label, info)
else:
query['label'] = ''
if info.get('jinv'):
j = clean_input(info['jinv'])
j = j.replace('+', '')
if not QQ_RE.match(j):
info['err'] = 'Error parsing input for the j-invariant. It needs to be a rational number.'
return search_input_error(info, bread)
query['jinv'] = j
for field in ['conductor', 'torsion', 'rank', 'sha_an']:
if info.get(field):
info[field] = clean_input(info[field])
ran = info[field]
ran = ran.replace('..', '-').replace(' ', '')
if not LIST_RE.match(ran):
names = {'conductor': 'conductor', 'torsion': 'torsion order', 'rank':
'rank', 'sha_an': 'analytic order of Ш'}
info['err'] = 'Error parsing input for the %s. It needs to be an integer (such as 5), a range of integers (such as 2-10 or 2..10), or a comma-separated list of these (such as 2,3,8 or 3-5, 7, 8-11).' % names[field]
return search_input_error(info, bread)
# Past input check
tmp = parse_range2(ran, field)
# work around syntax for $or
# we have to foil out multiple or conditions
if tmp[0] == '$or' and '$or' in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query['$or']]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
query[tmp[0]] = tmp[1]
# query[field] = parse_range2(info[field])
if 'optimal' in info and info['optimal'] == 'on':
# fails on 990h3
query['number'] = 1
if 'torsion_structure' in info and info['torsion_structure']:
info['torsion_structure'] = clean_input(info['torsion_structure'])
if not TORS_RE.match(info['torsion_structure']):
info['err'] = 'Error parsing input for the torsion structure. It needs to be one or more integers in square brackets, such as [6], [2,2], or [2,4]. Moreover, each integer should be bigger than 1, and each divides the next.'
return search_input_error(info, bread)
query['torsion_structure'] = [str(a) for a in parse_list(info['torsion_structure'])]
info['query'] = query
count_default = 100
if info.get('count'):
try:
count = int(info['count'])
except:
count = count_default
else:
count = count_default
info['count'] = count
start_default = 0
if info.get('start'):
try:
start = int(info['start'])
if(start < 0):
start += (1 - (start + 1) / count) * count
except:
start = start_default
else:
start = start_default
print query
cursor = lmfdb.base.getDBConnection().elliptic_curves.curves.find(query)
nres = cursor.count()
if(start >= nres):
start -= (1 + (start - nres) / count) * count
if(start < 0):
start = 0
res = cursor.sort([('conductor', ASCENDING), ('lmfdb_iso', ASCENDING), ('lmfdb_number', ASCENDING)
]).skip(start).limit(count)
info['curves'] = res
info['format_ainvs'] = format_ainvs
info['number'] = nres
info['start'] = start
if nres == 1:
info['report'] = 'unique match'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
credit = 'John Cremona'
t = 'Elliptic Curves'
return render_template("elliptic_curve/elliptic_curve_search.html", info=info, credit=credit, bread=bread, title=t)
def render_artin_representation_webpage(label):
if re.compile(r'^\d+$').match(label):
return artin_representation_search(**{'dimension': label})
bread = get_bread([(label, ' ')])
# label=dim.cond.nTt.indexcj, c is literal, j is index in conj class
# Should we have a big try around this to catch bad labels?
clean_label = clean_input(label)
if clean_label != label:
return redirect(url_for('.render_artin_representation_webpage', label=clean_label), 301)
try:
the_rep = ArtinRepresentation(label)
except:
flash(Markup("Error: <span style='color:black'>%s</span> is not the label of an Artin representation in the database." % (label)), "error")
return search_input_error({'err':''}, bread)
extra_data = {} # for testing?
extra_data['galois_knowl'] = group_display_knowl(5,3) # for testing?
#artin_logger.info("Found %s" % (the_rep._data))
title = "Artin Representation %s" % label
the_nf = the_rep.number_field_galois_group()
if the_rep.sign() == 0:
processed_root_number = "not computed"
else:
processed_root_number = str(the_rep.sign())
properties = [("Label", label),
("Dimension", str(the_rep.dimension())),
("Group", the_rep.group()),
("Conductor", "$" + the_rep.factored_conductor_latex() + "$"),
#("Bad primes", str(the_rep.bad_primes())),
("Root number", processed_root_number),
("Frobenius-Schur indicator", str(the_rep.indicator()))
]
friends = []
nf_url = the_nf.url_for()
if nf_url:
friends.append(("Artin Field", nf_url))
cc = the_rep.central_character()
if cc is not None:
if the_rep.dimension()==1:
if cc.order == 2:
cc_name = cc.symbol
else:
cc_name = cc.texname
friends.append(("Dirichlet character "+cc_name, url_for("characters.render_Dirichletwebpage", modulus=cc.modulus, number=cc.number)))
else:
detrep = the_rep.central_character_as_artin_rep()
friends.append(("Determinant representation "+detrep.label(), detrep.url_for()))
add_lfunction_friends(friends,label)
# once the L-functions are in the database, the link can always be shown
#if the_rep.dimension() <= 6:
if the_rep.dimension() == 1:
# Zeta is loaded differently
if cc.modulus == 1 and cc.number == 1:
friends.append(("L-function", url_for("l_functions.l_function_dirichlet_page", modulus=cc.modulus, number=cc.number)))
else:
# looking for Lhash dirichlet_L_modulus.number
mylhash = 'dirichlet_L_%d.%d'%(cc.modulus,cc.number)
lres = db.lfunc_instances.lucky({'Lhash': mylhash})
if lres is not None:
friends.append(("L-function", url_for("l_functions.l_function_dirichlet_page", modulus=cc.modulus, number=cc.number)))
# Dimension > 1
elif int(the_rep.conductor())**the_rep.dimension() <= 729000000000000:
friends.append(("L-function", url_for("l_functions.l_function_artin_page",
label=the_rep.label())))
info={}
return render_template("artin-representation-show.html", credit=tim_credit, support=support_credit, title=title, bread=bread, friends=friends, object=the_rep, properties2=properties, extra_data=extra_data, info=info, learnmore=learnmore_list())
def elliptic_curve_search(**args):
info = to_dict(args['data'])
if 'jump' in info:
label = info.get('label', '').replace(" ", "")
label_parts = label.split("-",1)
nf = label_parts[0]
label = label_parts[1]
return show_ecnf(nf,label)
query = {}
if 'conductor_norm' in info:
Nnorm = clean_input(info['conductor_norm'])
Nnorm = Nnorm.replace('..', '-').replace(' ', '')
tmp = parse_range2(Nnorm, 'conductor_norm')
if tmp[0] == '$or' and '$or' in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query['$or']]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
query[tmp[0]] = tmp[1]
if 'include_isogenous' in info and info['include_isogenous'] == 'off':
query['number'] = 1
if 'field' in info:
query['field_label'] = info['field']
info['query'] = query
# process count and start if not default:
count_default = 20
if info.get('count'):
try:
count = int(info['count'])
except:
count = count_default
else:
count = count_default
start_default = 0
if info.get('start'):
try:
start = int(info['start'])
if(start < 0):
start += (1 - (start + 1) / count) * count
except:
start = start_default
else:
start = start_default
# make the query and trim results according to start/count:
cursor = db_ecnf().find(query)
nres = cursor.count()
if(start >= nres):
start -= (1 + (start - nres) / count) * count
if(start < 0):
start = 0
res = cursor.sort([('field_label', ASC), ('conductor_norm', ASC), ('conductor_label', ASC), ('iso_label', ASC), ('number', ASC)]).skip(start).limit(count)
bread = []#[('Elliptic Curves over Number Fields', url_for(".elliptic_curve_search")), ]
bread = [('Elliptic Curves', url_for(".index")),
('Search Results', '.')]
res = list(res)
for e in res:
e['field_knowl'] = nf_display_knowl(e['field_label'], getDBConnection(), e['field_label'])
info['curves'] = res # [ECNF(e) for e in res]
info['number'] = nres
info['start'] = start
info['count'] = count
info['field_pretty'] = field_pretty
info['web_ainvs'] = web_ainvs
if nres == 1:
info['report'] = 'unique match'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
t = 'Elliptic Curve search results'
return render_template("ecnf-search-results.html", info=info, credit=ecnf_credit, bread=bread, title=t)
def parse_artin_orbit_label(label):
label = clean_input(label)
if ORBIT_RE.match(label):
return label
else:
raise ValueError
def render_group_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
label = label.replace('t', 'T')
C = base.getDBConnection()
data = C.transitivegroups.groups.find_one({'label': label})
if data is None:
bread = get_bread([("Search error", url_for('.search'))])
info['err'] = "Group " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
title = 'Galois Group:' + label
wgg = WebGaloisGroup.from_data(data)
n = data['n']
t = data['t']
data['yesno'] = yesno
order = data['order']
data['orderfac'] = latex(ZZ(order).factor())
orderfac = latex(ZZ(order).factor())
data['ordermsg'] = "$%s=%s$" % (order, latex(orderfac))
if order == 1:
data['ordermsg'] = "$1$"
if ZZ(order).is_prime():
data['ordermsg'] = "$%s$ (is prime)" % order
pgroup = len(ZZ(order).prime_factors()) < 2
if n == 1:
G = gap.SmallGroup(n, t)
else:
G = gap.TransitiveGroup(n, t)
if ZZ(order) < ZZ('10000000000'):
ctable = chartable(n, t)
else:
ctable = 'Group too large'
data['gens'] = generators(n, t)
if n == 1 and t == 1:
data['gens'] = 'None needed'
data['chartable'] = ctable
data['parity'] = "$%s$" % data['parity']
data['cclasses'] = conjclasses(G, n)
data['subinfo'] = subfield_display(C, n, data['subs'])
data['resolve'] = resolve_display(C, data['resolve'])
# if len(data['resolve']) == 0: data['resolve'] = 'None'
data['otherreps'] = wgg.otherrep_list()
query={'galois': bson.SON([('n', n), ('t', t)])}
C = base.getDBConnection()
intreps = C.transitivegroups.Gmodules.find({'n': n, 't': t})
# turn cursor into a list
intreps = [z for z in intreps]
data['int_reps'] = [galois_module_knowl(n, t, z['index'], C) for z in intreps]
data['int_reps_complete'] = int_reps_are_complete(intreps)
friends = []
one = C.numberfields.fields.find_one(query)
if one:
friends.append(('Number fields with this Galois group', url_for('number_fields.number_field_render_webpage')+"?galois_group=%dT%d" % (n, t) ))
prop2 = [
('Order:', '\(%s\)' % order),
('n:', '\(%s\)' % data['n']),
('Cyclic:', yesno(data['cyc'])),
('Abelian:', yesno(data['ab'])),
('Solvable:', yesno(data['solv'])),
('Primitive:', yesno(data['prim'])),
('$p$-group:', yesno(pgroup)),
('Name:', group_display_short(n, t, C)),
]
info.update(data)
bread = get_bread([(label, ' ')])
return render_template("gg-show-group.html", credit=GG_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends)
def number_field_search(**args):
info = to_dict(args)
info['learnmore'] = [('Global number field labels', url_for(".render_labels_page")), ('Galois group labels', url_for(".render_groups_page")), (Completename, url_for(".render_discriminants_page"))]
t = 'Global Number Field search results'
bread = [('Global Number Fields', url_for(".number_field_render_webpage")), ('Search results', ' ')]
# for k in info.keys():
# nf_logger.debug(str(k) + ' ---> ' + str(info[k]))
# nf_logger.debug('******************* '+ str(info['search']))
if 'natural' in info:
field_id = info['natural']
field_id_parsed = parse_field_string(info['natural'])
if FIELD_LABEL_RE.match(field_id_parsed):
field_id_parsed = split_label(field_id_parsed) # allows factored labels 11.11.11e20.1
return render_field_webpage({'label': field_id_parsed, 'label_orig': field_id})
query = {}
dlist = []
for field in ['galois_group', 'degree', 'signature', 'discriminant', 'class_number', 'class_group']:
if info.get(field):
info[field] = clean_input(info[field])
if field in ['class_group', 'signature']:
# different regex for the two types
if (field == 'signature' and PAIR_RE.match(info[field])) or (field == 'class_group' and IF_RE.match(info[field])):
query[field] = info[field][1:-1]
else:
name = 'class group' if field == 'class_group' else 'signature'
info['err'] = 'Error parsing input for %s. It needs to be a pair of integers in square brackets, such as [2,3] or [3,3]' % name
return search_input_error(info, bread)
else:
if field == 'galois_group':
try:
gcs = complete_group_codes(info[field])
if len(gcs) == 1:
query['galois'] = make_galois_pair(gcs[0][0], gcs[0][1])
# list(gcs[0])
if len(gcs) > 1:
query['galois'] = {'$in': [make_galois_pair(x[0], x[1]) for x in gcs]}
except NameError as code:
info['err'] = 'Error parsing input for Galois group: unknown group label %s. It needs to be a <a title = "Galois group labels" knowl="nf.galois_group.name">group label</a>, such as C5 or 5T1, or comma separated list of labels.' % code
return search_input_error(info, bread)
else: # not signature, class group, or galois group
ran = info[field]
ran = ran.replace('..', '-')
if LIST_RE.match(ran):
if field == 'discriminant':
# dlist will contain the disc conditions
# as sage ints
dlist = parse_discs(ran)
# now convert to a query
tmp = list_to_query(dlist)
# Two cases, could be a list of sign/inequalties
# or an '$or'
if len(tmp) == 1:
tmp = tmp[0]
else:
query[tmp[0][0]] = tmp[0][1]
tmp = tmp[1]
else:
tmp = parse_range2(ran, field)
# work around syntax for $or
# we have to foil out multiple or conditions
if tmp[0] == '$or' and '$or' in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query['$or']]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
query[tmp[0]] = tmp[1]
else:
name = re.sub('_', ' ', field)
info['err'] = 'Error parsing input for %s. It needs to be an integer (such as 5), a range of integers (such as 2-100 or 2..100), or a comma-separated list of these (such as 2,3,8 or 3-5, 7, 8-100).' % name
return search_input_error(info, bread)
if info.get('ur_primes'):
# now we want a list of strings, no spaces, which might be big ints
info['ur_primes'] = clean_input(info['ur_primes'])
if LIST_SIMPLE_RE.match(info['ur_primes']):
ur_primes = info['ur_primes'].split(',')
# Assuming this will be the only nor in the query
query['$nor'] = [{'ramps': x} for x in ur_primes]
else:
info['err'] = 'Error parsing input for unramified primes. It needs to be an integer (such as 5), or a comma-separated list of integers (such as 2,3,11).'
return search_input_error(info, bread)
if info.get('ram_primes'):
# now we want a list of strings, no spaces, which might be big ints
info['ram_primes'] = clean_input(info['ram_primes'])
if LIST_SIMPLE_RE.match(info['ram_primes']):
ram_primes = info['ram_primes'].split(',')
if str(info['ram_quantifier']) == 'some':
query['ramps'] = {'$all': ram_primes}
else:
query['ramps'] = ram_primes
else:
info['err'] = 'Error parsing input for ramified primes. It needs to be an integer (such as 5), or a comma-separated list of integers (such as 2,3,11).'
return search_input_error(info, bread)
count_default = 20
if info.get('count'):
try:
count = int(info['count'])
except:
count = count_default
info['count'] = count
else:
info['count'] = count_default
count = count_default
info['count'] = int(info['count'])
start_default = 0
if info.get('start'):
try:
start = int(info['start'])
if(start < 0):
start += (1 - (start + 1) / count) * count
except:
start = start_default
else:
start = start_default
if info.get('paging'):
try:
paging = int(info['paging'])
if paging == 0:
start = 0
except:
pass
C = base.getDBConnection()
# nf_logger.debug(query)
info['query'] = dict(query)
if 'lucky' in args:
one = C.numberfields.fields.find_one(query)
if one:
label = one['label']
return render_field_webpage({'label': label})
fields = C.numberfields.fields
res = fields.find(
query).sort([('degree', ASC), ('disc_abs_key', ASC), ('disc_sign', ASC), ('label', ASC)])
if 'download' in info and info['download'] != '0':
return download_search(info, res)
nres = res.count()
res = res.skip(start).limit(count)
if(start >= nres):
start -= (1 + (start - nres) / count) * count
if(start < 0):
start = 0
info['fields'] = res
info['number'] = nres
info['start'] = start
if nres == 1:
info['report'] = 'unique match'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
info['wnf'] = WebNumberField.from_data
return render_template("number_field_search.html", info=info, title=t, bread=bread)
def elliptic_curve_search(**args):
info = to_dict(args)
query = {}
bread = [('Elliptic Curves', url_for("ecnf.index")),
('$\Q$', url_for(".rational_elliptic_curves")),
('Search Results', '.')]
if 'SearchAgain' in args:
return rational_elliptic_curves()
if 'jump' in args:
label = info.get('label', '').replace(" ", "")
m = match_lmfdb_label(label)
if m:
try:
return by_ec_label(label)
except ValueError:
return elliptic_curve_jump_error(label, info, wellformed_label=True)
elif label.startswith("Cremona:"):
label = label[8:]
m = match_cremona_label(label)
if m:
try:
return by_ec_label(label)
except ValueError:
return elliptic_curve_jump_error(label, info, wellformed_label=True)
elif match_cremona_label(label):
return elliptic_curve_jump_error(label, info, cremona_label=True)
elif label:
# Try to parse a string like [1,0,3,2,4] as valid
# Weistrass coefficients:
lab = re.sub(r'\s','',label)
lab = re.sub(r'^\[','',lab)
lab = re.sub(r']$','',lab)
try:
labvec = lab.split(',')
labvec = [QQ(str(z)) for z in labvec] # Rationals allowed
E = EllipticCurve(labvec)
# Now we do have a valid curve over Q, but it might
# not be in the database.
ainvs = [str(c) for c in E.minimal_model().ainvs()]
data = db_ec().find_one({'ainvs': ainvs})
if data is None:
info['conductor'] = E.conductor()
return elliptic_curve_jump_error(label, info, missing_curve=True)
return by_ec_label(data['lmfdb_label'])
except (TypeError, ValueError, ArithmeticError):
return elliptic_curve_jump_error(label, info)
else:
query['label'] = ''
if info.get('jinv'):
j = clean_input(info['jinv'])
j = j.replace('+', '')
if not QQ_RE.match(j):
info['err'] = 'Error parsing input for the j-invariant. It needs to be a rational number.'
return search_input_error(info, bread)
query['jinv'] = str(QQ(j)) # to simplify e.g. 1728/1
for field in ['conductor', 'torsion', 'rank', 'sha']:
if info.get(field):
info[field] = clean_input(info[field])
ran = info[field]
ran = ran.replace('..', '-').replace(' ', '')
if not LIST_RE.match(ran):
names = {'conductor': 'conductor', 'torsion': 'torsion order', 'rank':
'rank', 'sha': 'analytic order of Ш'}
info['err'] = 'Error parsing input for the %s. It needs to be an integer (such as 25), a range of integers (such as 2-10 or 2..10), or a comma-separated list of these (such as 4,9,16 or 4-25, 81-121).' % names[field]
return search_input_error(info, bread)
# Past input check
tmp = parse_range2(ran, field)
# work around syntax for $or
# we have to foil out multiple or conditions
if tmp[0] == '$or' and '$or' in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query['$or']]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
query[tmp[0]] = tmp[1]
if 'optimal' in info and info['optimal'] == 'on':
# fails on 990h3
query['number'] = 1
if 'torsion_structure' in info and info['torsion_structure']:
res = parse_torsion_structure(info['torsion_structure'])
if 'Error' in res:
info['err'] = res
return search_input_error(info, bread)
#update info for repeat searches
info['torsion_structure'] = str(res).replace(' ','')
query['torsion_structure'] = [str(r) for r in res]
if info.get('surj_primes'):
info['surj_primes'] = clean_input(info['surj_primes'])
format_ok = LIST_POSINT_RE.match(info['surj_primes'])
if format_ok:
surj_primes = [int(p) for p in info['surj_primes'].split(',')]
format_ok = all([ZZ(p).is_prime(proof=False) for p in surj_primes])
if format_ok:
query['non-surjective_primes'] = {"$nin": surj_primes}
else:
info['err'] = 'Error parsing input for surjective primes. It needs to be a prime (such as 5), or a comma-separated list of primes (such as 2,3,11).'
return search_input_error(info, bread)
if info.get('nonsurj_primes'):
info['nonsurj_primes'] = clean_input(info['nonsurj_primes'])
format_ok = LIST_POSINT_RE.match(info['nonsurj_primes'])
if format_ok:
nonsurj_primes = [int(p) for p in info['nonsurj_primes'].split(',')]
format_ok = all([ZZ(p).is_prime(proof=False) for p in nonsurj_primes])
if format_ok:
if info['surj_quantifier'] == 'exactly':
nonsurj_primes.sort()
query['non-surjective_primes'] = nonsurj_primes
else:
if 'non-surjective_primes' in query:
query['non-surjective_primes'] = { "$nin": surj_primes, "$all": nonsurj_primes }
else:
query['non-surjective_primes'] = { "$all": nonsurj_primes }
else:
info['err'] = 'Error parsing input for nonsurjective primes. It needs to be a prime (such as 5), or a comma-separated list of primes (such as 2,3,11).'
return search_input_error(info, bread)
if 'download' in info and info['download'] != '0':
res = db_ec().find(query).sort([ ('conductor', ASCENDING), ('lmfdb_iso', ASCENDING), ('lmfdb_number', ASCENDING) ])
return download_search(info, res)
count_default = 100
if info.get('count'):
try:
count = int(info['count'])
except:
count = count_default
else:
count = count_default
info['count'] = count
start_default = 0
if info.get('start'):
try:
start = int(info['start'])
if(start < 0):
start += (1 - (start + 1) / count) * count
except:
start = start_default
else:
start = start_default
cursor = db_ec().find(query)
nres = cursor.count()
if(start >= nres):
start -= (1 + (start - nres) / count) * count
if(start < 0):
start = 0
res = cursor.sort([('conductor', ASCENDING), ('lmfdb_iso', ASCENDING), ('lmfdb_number', ASCENDING)
]).skip(start).limit(count)
info['curves'] = res
info['format_ainvs'] = format_ainvs
info['curve_url'] = lambda dbc: url_for(".by_triple_label", conductor=dbc['conductor'], iso_label=split_lmfdb_label(dbc['lmfdb_iso'])[1], number=dbc['lmfdb_number'])
info['iso_url'] = lambda dbc: url_for(".by_double_iso_label", conductor=dbc['conductor'], iso_label=split_lmfdb_label(dbc['lmfdb_iso'])[1])
info['number'] = nres
info['start'] = start
if nres == 1:
info['report'] = 'unique match'
elif nres == 2:
info['report'] = 'displaying both matches'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
credit = 'John Cremona'
if 'non-surjective_primes' in query:
credit += 'and Andrew Sutherland'
t = 'Elliptic Curves search results'
return render_template("search_results.html", info=info, credit=credit, bread=bread, title=t)
def by_label(label):
clean_label = clean_input(label)
if label != clean_label:
return redirect(url_for_label(label=clean_label), 301)
return render_field_webpage({'label': label})
def render_lattice_webpage(**args):
f = None
if 'label' in args:
lab = clean_input(args.get('label'))
if lab != args.get('label'):
return redirect(url_for('.render_lattice_webpage', label=lab), 301)
f = db.lat_lattices.lucky({'$or':[{'label': lab }, {'name': {'$contains': [lab]}}]})
if f is None:
t = "Integral Lattices Search Error"
bread = [('Lattices', url_for(".lattice_render_webpage"))]
flash(Markup("Error: <span style='color:black'>%s</span> is not a valid label or name for an integral lattice in the database." % (lab)),"error")
return render_template("lattice-error.html", title=t, properties=[], bread=bread, learnmore=learnmore_list())
info = {}
info.update(f)
info['friends'] = []
bread = [('Lattice', url_for(".lattice_render_webpage")), ('%s' % f['label'], ' ')]
credit = lattice_credit
info['dim']= int(f['dim'])
info['det']= int(f['det'])
info['level']=int(f['level'])
info['gram']=vect_to_matrix(f['gram'])
info['density']=str(f['density'])
info['hermite']=str(f['hermite'])
info['minimum']=int(f['minimum'])
info['kissing']=int(f['kissing'])
info['aut']=int(f['aut'])
if f['shortest']=="":
info['shortest']==f['shortest']
else:
if f['dim']==1:
info['shortest']=str(f['shortest']).strip('[').strip(']')
else:
if info['dim']*info['kissing']<100:
info['shortest']=[str([tuple(v)]).strip('[').strip(']').replace('),', '), ') for v in f['shortest']]
else:
max_vect_num=min(int(round(100/(info['dim']))), int(round(info['kissing']/2))-1);
info['shortest']=[str([tuple(f['shortest'][i])]).strip('[').strip(']').replace('),', '), ') for i in range(max_vect_num+1)]
info['all_shortest']="no"
info['download_shortest'] = [
(i, url_for(".render_lattice_webpage_download", label=info['label'], lang=i, obj='shortest_vectors')) for i in ['gp', 'magma','sage']]
if f['name']==['Leech']:
info['shortest']=[str([1,-2,-2,-2,2,-1,-1,3,3,0,0,2,2,-1,-1,-2,2,-2,-1,-1,0,0,-1,2]),
str([1,-2,-2,-2,2,-1,0,2,3,0,0,2,2,-1,-1,-2,2,-1,-1,-2,1,-1,-1,3]), str([1,-2,-2,-1,1,-1,-1,2,2,0,0,2,2,0,0,-2,2,-1,-1,-1,0,-1,-1,2])]
info['all_shortest']="no"
info['download_shortest'] = [
(i, url_for(".render_lattice_webpage_download", label=info['label'], lang=i, obj='shortest_vectors')) for i in ['gp', 'magma','sage']]
ncoeff=20
if f['theta_series'] != "":
coeff=[f['theta_series'][i] for i in range(ncoeff+1)]
info['theta_series']=my_latex(print_q_expansion(coeff))
info['theta_display'] = url_for(".theta_display", label=f['label'], number="")
info['class_number']=int(f['class_number'])
if f['dim']==1:
info['genus_reps']=str(f['genus_reps']).strip('[').strip(']')
else:
if info['dim']*info['class_number']<50:
info['genus_reps']=[vect_to_matrix(n) for n in f['genus_reps']]
else:
max_matrix_num=min(int(round(25/(info['dim']))), info['class_number']);
info['all_genus_rep']="no"
info['genus_reps']=[vect_to_matrix(f['genus_reps'][i]) for i in range(max_matrix_num+1)]
info['download_genus_reps'] = [
(i, url_for(".render_lattice_webpage_download", label=info['label'], lang=i, obj='genus_reps')) for i in ['gp', 'magma','sage']]
if f['name'] != "":
if f['name']==str(f['name']):
info['name']= str(f['name'])
else:
info['name']=str(", ".join(str(i) for i in f['name']))
else:
info['name'] == ""
info['comments']=str(f['comments'])
if 'Leech' in info['comments']: # no need to duplicate as it is in the name
info['comments'] = ''
if info['name'] == "":
t = "Integral Lattice %s" % info['label']
else:
t = "Integral Lattice "+info['label']+" ("+info['name']+")"
#This part code was for the dinamic knowl with comments, since the test is displayed this is redundant
# if info['name'] != "" or info['comments'] !="":
# info['knowl_args']= "name=%s&report=%s" %(info['name'], info['comments'].replace(' ', '-space-'))
info['properties'] = [
('Dimension', '%s' %info['dim']),
('Determinant', '%s' %info['det']),
('Level', '%s' %info['level'])]
if info['class_number'] == 0:
info['properties']=[('Class number', 'not available')]+info['properties']
else:
info['properties']=[('Class number', '%s' %info['class_number'])]+info['properties']
info['properties']=[('Label', '%s' % info['label'])]+info['properties']
if info['name'] != "" :
info['properties']=[('Name','%s' % info['name'] )]+info['properties']
# friends = [('L-series (not available)', ' ' ),('Half integral weight modular forms (not available)', ' ')]
return render_template("lattice-single.html", info=info, credit=credit, title=t, bread=bread, properties2=info['properties'], learnmore=learnmore_list(), KNOWL_ID="lattice.%s"%info['label'])
def render_hecke_algebras_webpage_l_adic(**args):
data = None
if 'orbit_label' in args and 'prime' in args:
lab = clean_input(args.get('orbit_label'))
if lab != args.get('orbit_label'):
base_lab=".".join([split(lab)[i] for i in [0,1,2]])
return redirect(url_for('.render_hecke_algebras_webpage', label=base_lab), 301)
try:
ell = int(args.get('prime'))
except ValueError:
base_lab=".".join([split(lab)[i] for i in [0,1,2]])
return redirect(url_for('.render_hecke_algebras_webpage', label=base_lab), 301)
data = db.hecke_ladic.lucky({'orbit_label': lab , 'ell': ell})
if data is None:
t = "Hecke Algebras Search Error"
bread = [('HeckeAlgebra', url_for(".hecke_algebras_render_webpage"))]
flash(Markup("Error: <span style='color:black'>%s</span> is not a valid label for the ℓ-adic information for an Hecke Algebra orbit in the database." % (lab)),"error")
return render_template("hecke_algebras-error.html", title=t, properties=[], bread=bread, learnmore=learnmore_list())
info = {}
info.update(data)
proj = ['index','orbit_label','ell','idempotent','field','structure','properties','operators']
res = list(db.hecke_ladic.search({'level': data['level'],'weight': data['weight'],'orbit_label': data['orbit_label'], 'ell': data['ell']}, proj))
for f in res:
if f['idempotent'] != "":
dim = len(sage_eval(f['idempotent']))
l_max = sage_eval(f['idempotent'])[0][0].ndigits()
if dim > 4 or l_max > 5:
f['idempotent_display'] = []
elif dim == 1:
f['idempotent_display'] = sage_eval(f['idempotent'])[0][0]
else:
f['idempotent_display'] = latex(matrix(sage_eval(f['idempotent'])))
else:
f['idempotent_display']=latex(matrix([[1]]))
del f['idempotent']
f['download_id'] = [(i, url_for(".render_hecke_algebras_webpage_ell_download", orbit_label=f['orbit_label'], index=f['index'], prime=f['ell'], lang=i, obj='idempotents')) for i in ['magma','sage']] # for 'gp' the code does not work, since p-adics are not implemented
field = f.pop('field')
if field is not None:
f['deg'] = field[1]
f['field_poly'] = field[2]
structure = f.pop('structure')
if structure is not None:
f['dim'] = structure[0]
f['num_gen'] = structure[1]
s2 = sage_eval(structure[2])
f['gens'] = [[int(s2.index(i)+1), str(i)] for i in s2]
f['rel'] = sage_eval(structure[3])
properties = f.pop('properties')
if properties is not None:
f['grading'] = properties[0]
f['gorenstein_def'] = properties[1]
f['gorenstein'] = "yes" if properties[1] == 0 else "no"
operators = f.pop('operators')
if operators is not None:
f['operators_mod_l'] = operators
f['num_hecke_op'] = len(operators)
f['size_op'] = size = sqrt(len(operators[0]))
if size > 4:
f['operators_mod_l_display'] = []
elif size == 1:
f['operators_mod_l_display'] = [[i+1, operators[i][0]] for i in range(10)]
else:
f['operators_mod_l_display'] = [[i+1, latex(matrix(size,size,operators[i]))] for i in range(5)]
f['download_op'] = [(lang, url_for(".render_hecke_algebras_webpage_ell_download", orbit_label=f['orbit_label'], index=f['index'], prime=f['ell'], lang=lang, obj='operators')) for lang in ['magma','sage']] # for 'gp' the code does not work
info['num_l_adic_orbits'] = len(res)
info['l_adic_orbits'] = res
info['level']=int(data['level'])
info['weight']= int(data['weight'])
info['base_lab']=".".join([split(data['orbit_label'])[i] for i in [0,1,2]])
info['orbit_label']= str(data['orbit_label'])
info['ell']=int(data['ell'])
bread = [('HeckeAlgebra', url_for(".hecke_algebras_render_webpage")), ('%s' % info['base_lab'], url_for('.render_hecke_algebras_webpage', label=info['base_lab'])), ('%s' % info['ell'], ' ')]
credit = hecke_algebras_credit
info['properties'] = [
('Level', '%s' %info['level']),
('Weight', '%s' %info['weight']),
('Characteristic', '%s' %info['ell']),
('Orbit label', '%s' %info['orbit_label'])]
info['friends'] = [('Modular form ' + info['base_lab'], url_for("cmf.by_url_space_label", level=info['level'], weight=info['weight'], char_orbit_label='a'))]
t = "%s-adic and mod %s Data for the Hecke Algebra Orbit %s" % (info['ell'], info['ell'], info['orbit_label'])
return render_template("hecke_algebras_l_adic-single.html", info=info, credit=credit, title=t, bread=bread, properties2=info['properties'], learnmore=learnmore_list(), friends=info['friends'], KNOWL_ID='hecke_algebra_l_adic.%s'%(info['orbit_label']))
def by_label(label):
clean_label = clean_input(label)
if clean_label != label:
return redirect(url_for('.by_label', label=clean_label), 301)
return render_group_webpage({'label': label})
def render_group_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
label = label.replace('t', 'T')
C = base.getDBConnection()
data = C.transitivegroups.groups.find_one({'label': label})
if data is None:
bread = get_bread([("Search error", url_for('.search'))])
info['err'] = "Group " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
title = 'Galois Group: ' + label
wgg = WebGaloisGroup.from_data(data)
n = data['n']
t = data['t']
data['yesno'] = yesno
order = data['order']
data['orderfac'] = latex(ZZ(order).factor())
orderfac = latex(ZZ(order).factor())
data['ordermsg'] = "$%s=%s$" % (order, latex(orderfac))
if order == 1:
data['ordermsg'] = "$1$"
if ZZ(order).is_prime():
data['ordermsg'] = "$%s$ (is prime)" % order
pgroup = len(ZZ(order).prime_factors()) < 2
if n == 1:
G = gap.SmallGroup(n, t)
else:
G = gap.TransitiveGroup(n, t)
if ZZ(order) < ZZ('10000000000'):
ctable = chartable(n, t)
else:
ctable = 'Group too large'
data['gens'] = generators(n, t)
if n == 1 and t == 1:
data['gens'] = 'None needed'
data['chartable'] = ctable
data['parity'] = "$%s$" % data['parity']
data['cclasses'] = conjclasses(G, n)
data['subinfo'] = subfield_display(C, n, data['subs'])
data['resolve'] = resolve_display(C, data['resolve'])
data['otherreps'] = wgg.otherrep_list()
if len(data['otherreps']) == 0:
data['otherreps']="There is no other low degree representation."
query={'galois': bson.SON([('n', n), ('t', t)])}
C = base.getDBConnection()
intreps = C.transitivegroups.Gmodules.find({'n': n, 't': t}).sort('index', pymongo.ASCENDING)
# turn cursor into a list
intreps = [z for z in intreps]
if len(intreps) > 0:
data['int_rep_classes'] = [str(z[0]) for z in intreps[0]['gens']]
for onerep in intreps:
onerep['gens']=[list_to_latex_matrix(z[1]) for z in onerep['gens']]
data['int_reps'] = intreps
data['int_reps_complete'] = int_reps_are_complete(intreps)
dcq = data['moddecompuniq']
if dcq[0] == 0:
data['decompunique'] = 0
else:
data['decompunique'] = dcq[0]
data['isoms'] = [[mult2mult(z[0]), mult2mult(z[1])] for z in dcq[1]]
data['isoms'] = [[modules2string(n,t,z[0]), modules2string(n,t,z[1])] for z in data['isoms']]
print dcq[1]
print data['isoms']
friends = []
one = C.numberfields.fields.find_one(query)
if one:
friends.append(('Number fields with this Galois group', url_for('number_fields.number_field_render_webpage')+"?galois_group=%dT%d" % (n, t) ))
prop2 = [
('Order:', '\(%s\)' % order),
('n:', '\(%s\)' % data['n']),
('Cyclic:', yesno(data['cyc'])),
('Abelian:', yesno(data['ab'])),
('Solvable:', yesno(data['solv'])),
('Primitive:', yesno(data['prim'])),
('$p$-group:', yesno(pgroup)),
]
pretty = group_display_pretty(n,t,C)
if len(pretty)>0:
prop2.extend([('Group:', pretty)])
info['pretty_name'] = pretty
data['name'] = re.sub(r'_(\d+)',r'_{\1}',data['name'])
data['name'] = re.sub(r'\^(\d+)',r'^{\1}',data['name'])
info.update(data)
bread = get_bread([(label, ' ')])
return render_template("gg-show-group.html", credit=GG_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends)
def by_label(label):
clean_label = clean_input(label)
if label != clean_label:
return redirect(url_for('.by_label',label=clean_label), 301)
return render_field_webpage({'label': label})
def render_artin_representation_webpage(label):
if re.compile(r'^\d+$').match(label):
return artin_representation_search(**{'dimension': label})
bread = get_bread([(label, ' ')])
# label=dim.cond.nTt.indexcj, c is literal, j is index in conj class
# Should we have a big try around this to catch bad labels?
clean_label = clean_input(label)
if clean_label != label:
return redirect(
url_for('.render_artin_representation_webpage', label=clean_label),
301)
try:
the_rep = ArtinRepresentation(label)
except:
try:
newlabel = parse_artin_label(label)
flash_error("Artin representation %s is not in database", newlabel)
return redirect(url_for(".index"))
except ValueError:
flash_error(
"%s is not in a valid form for an Artin representation label",
label)
return redirect(url_for(".index"))
extra_data = {} # for testing?
extra_data['galois_knowl'] = group_display_knowl(5, 3) # for testing?
#artin_logger.info("Found %s" % (the_rep._data))
title = "Artin Representation %s" % label
the_nf = the_rep.number_field_galois_group()
if the_rep.sign() == 0:
processed_root_number = "not computed"
else:
processed_root_number = str(the_rep.sign())
properties = [
("Label", label),
("Dimension", str(the_rep.dimension())),
("Group", the_rep.group()),
("Conductor", "$" + the_rep.factored_conductor_latex() + "$"),
#("Bad primes", str(the_rep.bad_primes())),
("Root number", processed_root_number),
("Frobenius-Schur indicator", str(the_rep.indicator()))
]
friends = []
nf_url = the_nf.url_for()
if nf_url:
friends.append(("Artin Field", nf_url))
cc = the_rep.central_character()
if cc is not None:
if the_rep.dimension() == 1:
if cc.order == 2:
cc_name = cc.symbol
else:
cc_name = cc.texname
friends.append(("Dirichlet character " + cc_name,
url_for("characters.render_Dirichletwebpage",
modulus=cc.modulus,
number=cc.number)))
else:
detrep = the_rep.central_character_as_artin_rep()
friends.append(("Determinant representation " + detrep.label(),
detrep.url_for()))
add_lfunction_friends(friends, label)
# once the L-functions are in the database, the link can always be shown
#if the_rep.dimension() <= 6:
if the_rep.dimension() == 1:
# Zeta is loaded differently
if cc.modulus == 1 and cc.number == 1:
friends.append(("L-function",
url_for("l_functions.l_function_dirichlet_page",
modulus=cc.modulus,
number=cc.number)))
else:
# looking for Lhash dirichlet_L_modulus.number
mylhash = 'dirichlet_L_%d.%d' % (cc.modulus, cc.number)
lres = db.lfunc_instances.lucky({'Lhash': mylhash})
if lres is not None:
friends.append(
("L-function",
url_for("l_functions.l_function_dirichlet_page",
modulus=cc.modulus,
number=cc.number)))
# Dimension > 1
elif int(the_rep.conductor())**the_rep.dimension() <= 729000000000000:
friends.append(("L-function",
url_for("l_functions.l_function_artin_page",
label=the_rep.label())))
info = {}
return render_template("artin-representation-show.html",
credit=tim_credit,
support=support_credit,
title=title,
bread=bread,
friends=friends,
object=the_rep,
properties2=properties,
extra_data=extra_data,
info=info,
learnmore=learnmore_list())
def render_field_webpage(args):
data = None
info = {}
bread = bread_prefix()
# This function should not be called unless label is set.
label = clean_input(args['label'])
nf = WebNumberField(label)
data = {}
if nf.is_null():
if re.match(r'^\d+\.\d+\.\d+\.\d+$', label):
flash_error("Number field %s was not found in the database.",
label)
else:
flash_error("%s is not a valid label for a number field.", label)
return redirect(url_for(".number_field_render_webpage"))
info['wnf'] = nf
data['degree'] = nf.degree()
data['class_number'] = nf.class_number_latex()
ram_primes = nf.ramified_primes()
t = nf.galois_t()
n = nf.degree()
data['is_galois'] = nf.is_galois()
data['autstring'] = r'\Gal' if data['is_galois'] else r'\Aut'
data['is_abelian'] = nf.is_abelian()
if nf.is_abelian():
conductor = nf.conductor()
data['conductor'] = conductor
dirichlet_chars = nf.dirichlet_group()
if dirichlet_chars:
data['dirichlet_group'] = [
r'<a href = "%s">$\chi_{%s}(%s,·)$</a>' %
(url_for('characters.render_Dirichletwebpage',
modulus=data['conductor'],
number=j), data['conductor'], j)
for j in dirichlet_chars
]
if len(data['dirichlet_group']) == 1:
data[
'dirichlet_group'] = r'<span style="white-space:nowrap">$\lbrace$' + data[
'dirichlet_group'][0] + r'$\rbrace$</span>'
else:
data['dirichlet_group'] = r'$\lbrace$' + ', '.join(
data['dirichlet_group']
[:-1]) + '<span style="white-space:nowrap">' + data[
'dirichlet_group'][-1] + r'$\rbrace$</span>'
if data['conductor'].is_prime() or data['conductor'] == 1:
data['conductor'] = r"\(%s\)" % str(data['conductor'])
else:
factored_conductor = factor_base_factor(data['conductor'],
ram_primes)
factored_conductor = factor_base_factorization_latex(
factored_conductor)
data['conductor'] = r"\(%s=%s\)" % (str(
data['conductor']), factored_conductor)
data['galois_group'] = group_pretty_and_nTj(n, t, True)
data['auts'] = db.gps_transitive.lookup(r'{}T{}'.format(n, t))['auts']
data['cclasses'] = cclasses_display_knowl(n, t)
data['character_table'] = character_table_display_knowl(n, t)
data['class_group'] = nf.class_group()
data['class_group_invs'] = nf.class_group_invariants()
data['signature'] = nf.signature()
data['coefficients'] = nf.coeffs()
nf.make_code_snippets()
D = nf.disc()
data['disc_factor'] = nf.disc_factored_latex()
if D.abs().is_prime() or D == 1:
data['discriminant'] = bigint_knowl(D, cutoff=60, sides=3)
else:
data['discriminant'] = bigint_knowl(
D, cutoff=60,
sides=3) + r"\(\medspace = %s\)" % data['disc_factor']
if nf.frobs():
data['frob_data'], data['seeram'] = see_frobs(nf.frobs())
else: # fallback in case we haven't computed them in a case
data['frob_data'], data['seeram'] = frobs(nf)
# This could put commas in the rd, we don't want to trigger spaces
data['rd'] = ('$%s$' % fixed_prec(nf.rd(), 2)).replace(',', '{,}')
# Bad prime information
npr = len(ram_primes)
ramified_algebras_data = nf.ramified_algebras_data()
if isinstance(ramified_algebras_data, str):
loc_alg = ''
else:
# [label, latex, e, f, c, gal]
loc_alg = ''
for j in range(npr):
if ramified_algebras_data[j] is None:
loc_alg += '<tr><td>%s<td colspan="7">Data not computed' % str(
ram_primes[j]).rstrip('L')
else:
from lmfdb.local_fields.main import show_slope_content
mydat = ramified_algebras_data[j]
p = ram_primes[j]
loc_alg += '<tr><td rowspan="%d">$%s$</td>' % (len(mydat),
str(p))
mm = mydat[0]
myurl = url_for('local_fields.by_label', label=mm[0])
lab = mm[0]
if mm[3] * mm[2] == 1:
lab = r'$\Q_{%s}$' % str(p)
loc_alg += '<td><a href="%s">%s</a><td>$%s$<td>$%d$<td>$%d$<td>$%d$<td>%s<td>$%s$' % (
myurl, lab, mm[1], mm[2], mm[3], mm[4], mm[5],
show_slope_content(mm[8], mm[6], mm[7]))
for mm in mydat[1:]:
lab = mm[0]
myurl = url_for('local_fields.by_label', label=lab)
if mm[3] * mm[2] == 1:
lab = r'$\Q_{%s}$' % str(p)
loc_alg += '<tr><td><a href="%s">%s</a><td>$%s$<td>$%d$<td>$%d$<td>$%d$<td>%s<td>$%s$' % (
myurl, lab, mm[1], mm[2], mm[3], mm[4], mm[5],
show_slope_content(mm[8], mm[6], mm[7]))
loc_alg += '</tbody></table>'
ram_primes = str(ram_primes)[1:-1]
# Get rid of python L for big numbers
ram_primes = ram_primes.replace('L', '')
if not ram_primes:
ram_primes = r'\textrm{None}'
data['phrase'] = group_phrase(n, t)
zk = nf.zk()
Ra = PolynomialRing(QQ, 'a')
zk = [latex(Ra(x)) for x in zk]
zk = ['$%s$' % x for x in zk]
zk = ', '.join(zk)
grh_label = '<small>(<a title="assuming GRH" knowl="nf.assuming_grh">assuming GRH</a>)</small>' if nf.used_grh(
) else ''
# Short version for properties
grh_lab = nf.short_grh_string()
if 'computed' in str(data['class_number']):
grh_lab = ''
grh_label = ''
pretty_label = field_pretty(label)
if label != pretty_label:
pretty_label = "%s: %s" % (label, pretty_label)
info.update(data)
rootofunity = '%s (order $%d$)' % (nf.root_of_1_gen(),
nf.root_of_1_order())
info.update({
'label': pretty_label,
'label_raw': label,
'polynomial': web_latex(nf.poly()),
'ram_primes': ram_primes,
'integral_basis': zk,
'regulator': web_latex(nf.regulator()),
'unit_rank': nf.unit_rank(),
'root_of_unity': rootofunity,
'fund_units': nf.units_safe(),
'cnf': nf.cnf(),
'grh_label': grh_label,
'loc_alg': loc_alg
})
bread.append(('%s' % nf_label_pretty(info['label_raw']), ' '))
info['downloads_visible'] = True
info['downloads'] = [('worksheet', '/')]
info['friends'] = []
if nf.can_class_number():
# hide ones that take a lond time to compute on the fly
# note that the first degree 4 number field missed the zero of the zeta function
if abs(D**n) < 50000000:
info['friends'].append(('L-function', "/L/NumberField/%s" % label))
info['friends'].append(('Galois group', "/GaloisGroup/%dT%d" % (n, t)))
if 'dirichlet_group' in info:
info['friends'].append(('Dirichlet character group',
url_for("characters.dirichlet_group_table",
modulus=int(conductor),
char_number_list=','.join(
str(a) for a in dirichlet_chars),
poly=info['polynomial'])))
resinfo = []
galois_closure = nf.galois_closure()
if galois_closure[0] > 0:
if galois_closure[1]:
resinfo.append(('gc', galois_closure[1]))
if galois_closure[2]:
info['friends'].append(('Galois closure',
url_for(".by_label",
label=galois_closure[2][0])))
else:
resinfo.append(('gc', [dnc]))
sextic_twins = nf.sextic_twin()
if sextic_twins[0] > 0:
if sextic_twins[1]:
resinfo.append(('sex', r' $\times$ '.join(sextic_twins[1])))
else:
resinfo.append(('sex', dnc))
siblings = nf.siblings()
# [degsib list, label list]
# first is list of [deg, num expected, list of knowls]
if siblings[0]:
for sibdeg in siblings[0]:
if not sibdeg[2]:
sibdeg[2] = dnc
else:
nsibs = len(sibdeg[2])
sibdeg[2] = ', '.join(sibdeg[2])
if nsibs < sibdeg[1]:
sibdeg[2] += ', some ' + dnc
resinfo.append(('sib', siblings[0]))
for lab in siblings[1]:
if lab:
labparts = lab.split('.')
info['friends'].append(("Degree %s sibling" % labparts[0],
url_for(".by_label", label=lab)))
arith_equiv = nf.arith_equiv()
if arith_equiv[0] > 0:
if arith_equiv[1]:
resinfo.append(
('ae', ', '.join(arith_equiv[1]), len(arith_equiv[1])))
for aelab in arith_equiv[2]:
info['friends'].append(('Arithmetically equivalent sibling',
url_for(".by_label", label=aelab)))
else:
resinfo.append(('ae', dnc, len(arith_equiv[1])))
info['resinfo'] = resinfo
learnmore = learnmore_list()
title = "Number field %s" % info['label']
if npr == 1:
primes = 'prime'
else:
primes = 'primes'
if len(ram_primes) > 30:
ram_primes = 'see page'
else:
ram_primes = '$%s$' % ram_primes
properties = [('Label', nf_label_pretty(label)),
('Degree', prop_int_pretty(data['degree'])),
('Signature', '$%s$' % data['signature']),
('Discriminant', prop_int_pretty(D)),
('Root discriminant', '%s' % data['rd']),
('Ramified ' + primes + '', ram_primes),
('Class number', '%s %s' % (data['class_number'], grh_lab)),
('Class group',
'%s %s' % (data['class_group_invs'], grh_lab)),
('Galois group', group_pretty_and_nTj(data['degree'], t))]
downloads = [('Stored data to gp',
url_for('.nf_download', nf=label, download_type='data'))]
for lang in [["Magma", "magma"], ["SageMath", "sage"], ["Pari/GP", "gp"]]:
downloads.append(('Download {} code'.format(lang[0]),
url_for(".nf_download",
nf=label,
download_type=lang[1])))
from lmfdb.artin_representations.math_classes import NumberFieldGaloisGroup
from lmfdb.artin_representations.math_classes import artin_label_pretty
try:
info["tim_number_field"] = NumberFieldGaloisGroup(nf._data['coeffs'])
arts = [
z.label()
for z in info["tim_number_field"].artin_representations()
]
#print arts
for ar in arts:
info['friends'].append((
'Artin representation ' + artin_label_pretty(ar),
url_for(
"artin_representations.render_artin_representation_webpage",
label=ar)))
v = nf.factor_perm_repn(info["tim_number_field"])
def dopow(m):
if m == 0:
return ''
if m == 1:
return '*'
return '*<sup>%d</sup>' % m
info["mydecomp"] = [dopow(x) for x in v]
except AttributeError:
pass
return render_template("nf-show-field.html",
properties=properties,
credit=NF_credit,
title=title,
bread=bread,
code=nf.code,
friends=info.pop('friends'),
downloads=downloads,
learnmore=learnmore,
info=info,
KNOWL_ID="nf.%s" % label)
def render_artin_representation_webpage(label):
if re.compile(r'^\d+$').match(label):
return artin_representation_search(**{
'dimension': label,
'search_array': ArtinSearchArray()
})
# label=dim.cond.nTt.indexcj, c is literal, j is index in conj class
# Should we have a big try around this to catch bad labels?
clean_label = clean_input(label)
if clean_label != label:
return redirect(
url_for('.render_artin_representation_webpage', label=clean_label),
301)
# We could have a single representation or a Galois orbit
case = parse_any(label)
if case[0] == 'malformed':
try:
raise ValueError
except:
flash_error(
"%s is not in a valid form for the label for an Artin representation or a Galois orbit of Artin representations",
label)
return redirect(url_for(".index"))
# Do this twice to customize error messages
newlabel = case[1]
case = case[0]
if case == 'rep':
try:
the_rep = ArtinRepresentation(newlabel)
except:
newlabel = parse_artin_label(label)
flash_error("Artin representation %s is not in database", label)
return redirect(url_for(".index"))
else: # it is an orbit
try:
the_rep = ArtinRepresentation(newlabel + '.a')
except:
newlabel = parse_artin_orbit_label(newlabel)
flash_error(
"Galois orbit of Artin representations %s is not in database",
label)
return redirect(url_for(".index"))
# in this case we want all characters
num_conj = the_rep.galois_conjugacy_size()
allchars = [
ArtinRepresentation(newlabel + '.' +
num2letters(j)).character_formatted()
for j in range(1, num_conj + 1)
]
label = newlabel
bread = get_bread([(label, ' ')])
#artin_logger.info("Found %s" % (the_rep._data))
if case == 'rep':
title = "Artin representation %s" % label
else:
title = "Galois orbit of Artin representations %s" % label
the_nf = the_rep.number_field_galois_group()
if the_rep.sign() == 0:
processed_root_number = "not computed"
else:
processed_root_number = str(the_rep.sign())
properties = [("Label", label), ("Dimension", str(the_rep.dimension())),
("Group", the_rep.group()),
("Conductor", "$" + the_rep.factored_conductor_latex() + "$")
]
if case == 'rep':
properties.append(("Root number", processed_root_number))
properties.append(("Frobenius-Schur indicator", str(the_rep.indicator())))
friends = []
wnf = None
nf_url = the_nf.url_for()
if nf_url:
friends.append(("Artin field", nf_url))
wnf = the_nf.wnf()
proj_nf = WebNumberField.from_coeffs(the_rep._data['Proj_Polynomial'])
if proj_nf:
friends.append(
("Projective Artin field",
str(url_for("number_fields.by_label",
label=proj_nf.get_label()))))
if case == 'rep':
cc = the_rep.central_character()
if cc is not None:
if the_rep.dimension() == 1:
if cc.order == 2:
cc_name = cc.symbol
else:
cc_name = cc.texname
friends.append(("Dirichlet character " + cc_name,
url_for("characters.render_Dirichletwebpage",
modulus=cc.modulus,
number=cc.number)))
else:
detrep = the_rep.central_character_as_artin_rep()
friends.append(("Determinant representation " + detrep.label(),
detrep.url_for()))
add_lfunction_friends(friends, label)
# once the L-functions are in the database, the link can always be shown
#if the_rep.dimension() <= 6:
if the_rep.dimension() == 1:
# Zeta is loaded differently
if cc.modulus == 1 and cc.number == 1:
friends.append(
("L-function",
url_for("l_functions.l_function_dirichlet_page",
modulus=cc.modulus,
number=cc.number)))
else:
# looking for Lhash dirichlet_L_modulus.number
mylhash = 'dirichlet_L_%d.%d' % (cc.modulus, cc.number)
lres = db.lfunc_instances.lucky({'Lhash': mylhash})
if lres is not None:
friends.append(
("L-function",
url_for("l_functions.l_function_dirichlet_page",
modulus=cc.modulus,
number=cc.number)))
# Dimension > 1
elif int(the_rep.conductor())**the_rep.dimension() <= 729000000000000:
friends.append(("L-function",
url_for("l_functions.l_function_artin_page",
label=the_rep.label())))
orblabel = re.sub(r'\.[a-z]+$', '', label)
friends.append(("Galois orbit " + orblabel,
url_for(".render_artin_representation_webpage",
label=orblabel)))
else:
add_lfunction_friends(friends, label)
friends.append(("L-function",
url_for("l_functions.l_function_artin_page",
label=the_rep.label())))
for j in range(1, 1 + the_rep.galois_conjugacy_size()):
newlabel = label + '.' + num2letters(j)
friends.append(("Artin representation " + newlabel,
url_for(".render_artin_representation_webpage",
label=newlabel)))
info = {} # for testing
if case == 'rep':
return render_template("artin-representation-show.html",
credit=tim_credit,
support=support_credit,
title=title,
bread=bread,
friends=friends,
object=the_rep,
cycle_string=cycle_string,
wnf=wnf,
properties=properties,
info=info,
learnmore=learnmore_list())
# else we have an orbit
return render_template("artin-representation-galois-orbit.html",
credit=tim_credit,
support=support_credit,
title=title,
bread=bread,
allchars=allchars,
friends=friends,
object=the_rep,
cycle_string=cycle_string,
wnf=wnf,
properties=properties,
info=info,
learnmore=learnmore_list())
def hgm_search(**args):
info = to_dict(args)
bread = get_bread([("Search results", url_for('.search'))])
C = base.getDBConnection()
query = {}
if 'jump_to' in info:
return render_hgm_webpage({'label': info['jump_to']})
family_search = False
if info.get('Submit Family') or info.get('family'):
family_search = True
# generic, irreducible not in DB yet
for param in [
'A', 'B', 'hodge', 'a2', 'b2', 'a3', 'b3', 'a5', 'b5', 'a7', 'b7'
]:
if info.get(param):
info[param] = clean_input(info[param])
if IF_RE.match(info[param]):
query[param] = parse_list(info[param])
query[param].sort()
else:
name = param
if field == 'hodge':
name = 'Hodge vector'
info[
'err'] = 'Error parsing input for %s. It needs to be a list of integers in square brackets, such as [2,3] or [1,1,1]' % name
return search_input_error(info, bread)
if info.get('t') and not family_search:
info['t'] = clean_input(info['t'])
try:
tsage = QQ(str(info['t']))
tlist = [int(tsage.numerator()), int(tsage.denominator())]
query['t'] = tlist
except:
info[
'err'] = 'Error parsing input for t. It needs to be a rational number, such as 2/3 or -3'
# sign can only be 1, -1, +1
if info.get('sign') and not family_search:
sign = info['sign']
sign = re.sub(r'\s', '', sign)
sign = clean_input(sign)
if sign == '+1':
sign = '1'
if not (sign == '1' or sign == '-1'):
info[
'err'] = 'Error parsing input %s for sign. It needs to be 1 or -1' % sign
return search_input_error(info, bread)
query['sign'] = int(sign)
for param in ['degree', 'weight', 'conductor']:
# We don't look at conductor in family searches
if info.get(param) and not (param == 'conductor' and family_search):
if param == 'conductor':
cond = info['conductor']
try:
cond = re.sub(r'(\d)\s+(\d)', r'\1 * \2',
cond) # implicit multiplication of numbers
cond = cond.replace(r'..', r'-') # all ranges use -
cond = re.sub(r'[a..zA..Z]', '', cond)
cond = clean_input(cond)
tmp = parse_range2(cond, 'cond', myZZ)
except:
info[
'err'] = 'Error parsing input for conductor. It needs to be an integer (e.g., 8), a range of integers (e.g. 10-100), or a list of such (e.g., 5,7,8,10-100). Integers may be given in factored form (e.g. 2^5 3^2) %s' % cond
return search_input_error(info, bread)
else: # not conductor
info[param] = clean_input(info[param])
ran = info[param]
ran = ran.replace(r'..', r'-')
if LIST_RE.match(ran):
tmp = parse_range2(ran, param)
else:
names = {'weight': 'weight', 'degree': 'degree'}
info[
'err'] = 'Error parsing input for the %s. It needs to be an integer (such as 5), a range of integers (such as 2-10 or 2..10), or a comma-separated list of these (such as 2,3,8 or 3-5, 7, 8-11).' % names[
param]
return search_input_error(info, bread)
# work around syntax for $or
# we have to foil out multiple or conditions
if tmp[0] == '$or' and '$or' in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query['$or']]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
query[tmp[0]] = tmp[1]
#print query
count_default = 20
if info.get('count'):
try:
count = int(info['count'])
except:
count = count_default
else:
count = count_default
info['count'] = count
start_default = 0
if info.get('start'):
try:
start = int(info['start'])
if (start < 0):
start += (1 - (start + 1) / count) * count
except:
start = start_default
else:
start = start_default
if info.get('paging'):
try:
paging = int(info['paging'])
if paging == 0:
start = 0
except:
pass
# logger.debug(query)
if family_search:
res = C.hgm.families.find(query).sort([('label', pymongo.ASCENDING)])
else:
res = C.hgm.motives.find(query).sort([('cond', pymongo.ASCENDING),
('label', pymongo.ASCENDING)])
nres = res.count()
res = res.skip(start).limit(count)
if (start >= nres):
start -= (1 + (start - nres) / count) * count
if (start < 0):
start = 0
info['motives'] = res
info['number'] = nres
info['start'] = start
if nres == 1:
info['report'] = 'unique match'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (
start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
info['make_label'] = make_abt_label
info['make_t_label'] = make_t_label
info['ab_label'] = ab_label
info['display_t'] = display_t
info['family'] = family_search
info['factorint'] = factorint
if family_search:
return render_template(
"hgm-search.html",
info=info,
title="Hypergeometric Family over $\Q$ Search Result",
bread=bread,
credit=HGM_credit)
else:
return render_template(
"hgm-search.html",
info=info,
title="Hypergeometric Motive over $\Q$ Search Result",
bread=bread,
credit=HGM_credit)
def render_field_webpage(args):
data = None
C = base.getDBConnection()
info = {}
bread = [('Global Number Fields', url_for(".number_field_render_webpage"))]
if 'label' in args:
label = clean_input(args['label'])
nf = WebNumberField(label)
data = {}
if nf.is_null():
bread.append(('Search results', ' '))
label2 = re.sub(r'[<>]', '', args['label'])
if 'You need to enter a field' in label2:
info['err'] = label2
else:
info['err'] = 'No such field: %s in the database' % label2
info['label'] = args['label_orig'] if 'label_orig' in args else args['label']
return search_input_error(info, bread)
info['wnf'] = nf
data['degree'] = nf.degree()
data['class_number'] = nf.class_number()
t = nf.galois_t()
n = nf.degree()
data['is_galois'] = nf.is_galois()
data['is_abelian'] = nf.is_abelian()
if nf.is_abelian():
conductor = nf.conductor()
data['conductor'] = conductor
dirichlet_chars = nf.dirichlet_group()
data['dirichlet_group'] = ['<a href = "%s">$\chi_{%s}(%s,·)$</a>' % (url_for("render_Character", arg1=data['conductor'], arg2=j), data['conductor'], j) for j in dirichlet_chars]
data['dirichlet_group'] = r'$\lbrace$' + ', '.join(data['dirichlet_group']) + r'$\rbrace$'
if data['conductor'].is_prime() or data['conductor'] == 1:
data['conductor'] = "\(%s\)" % str(data['conductor'])
else:
data['conductor'] = "\(%s=%s\)" % (str(data['conductor']), latex(data['conductor'].factor()))
data['galois_group'] = group_display_knowl(n, t, C)
data['cclasses'] = cclasses_display_knowl(n, t, C)
data['character_table'] = character_table_display_knowl(n, t, C)
data['class_group'] = nf.class_group()
data['class_group_invs'] = nf.class_group_invariants()
data['signature'] = nf.signature()
D = nf.disc()
ram_primes = D.prime_factors()
data['disc_factor'] = nf.disc_factored_latex()
if D.abs().is_prime() or D == 1:
data['discriminant'] = "\(%s\)" % str(D)
else:
data['discriminant'] = "\(%s=%s\)" % (str(D), data['disc_factor'])
npr = len(ram_primes)
ram_primes = str(ram_primes)[1:-1]
if ram_primes == '':
ram_primes = r'\textrm{None}'
data['frob_data'], data['seeram'] = frobs(nf.K())
data['phrase'] = group_phrase(n, t, C)
zk = pari(nf.K()).nf_subst('a')
zk = list(zk.nf_get_zk())
Ra = PolynomialRing(QQ, 'a')
zk = [latex(Ra(x)) for x in zk]
zk = ['$%s$' % x for x in zk]
zk = ', '.join(zk)
grh_label = '<small>(<a title="assuming GRH" knowl="nf.assuming_grh">assuming GRH</a>)</small>' if nf.used_grh() else ''
# Short version for properties
grh_lab = nf.short_grh_string()
pretty_label = field_pretty(label)
if label != pretty_label:
pretty_label = "%s: %s" % (label, pretty_label)
info.update(data)
info.update({
'label': pretty_label,
'label_raw': label,
'polynomial': web_latex_split_on_pm(nf.K().defining_polynomial()),
'ram_primes': ram_primes,
'integral_basis': zk,
'regulator': web_latex(nf.regulator()),
'unit_rank': nf.unit_rank(),
'root_of_unity': web_latex(nf.K().primitive_root_of_unity()),
'fund_units': nf.units(),
'grh_label': grh_label
})
bread.append(('%s' % info['label_raw'], ' '))
info['downloads_visible'] = True
info['downloads'] = [('worksheet', '/')]
info['friends'] = []
if nf.can_class_number():
info['friends'].append(('L-function', "/L/NumberField/%s" % label))
info['friends'].append(('Galois group', "/GaloisGroup/%dT%d" % (n, t)))
if 'dirichlet_group' in info:
info['friends'].append(('Dirichlet group', url_for("dirichlet_group_table",
modulus=int(conductor),
char_number_list=','.join(
[str(a) for a in dirichlet_chars]),
poly=info['polynomial'])))
info['learnmore'] = [('Global number field labels', url_for(
".render_labels_page")), (Completename, url_for(".render_discriminants_page"))]
# With Galois group labels, probably not needed here
# info['learnmore'] = [('Global number field labels',
# url_for(".render_labels_page")), ('Galois group
# labels',url_for(".render_groups_page")),
# (Completename,url_for(".render_discriminants_page"))]
title = "Global Number Field %s" % info['label']
if npr == 1:
primes = 'prime'
else:
primes = 'primes'
properties2 = [('Degree:', '%s' % data['degree']),
('Signature:', '$%s$' % data['signature']),
('Discriminant:', '$%s$' % data['disc_factor']),
('Ramified ' + primes + ':', '$%s$' % ram_primes),
('Class number:', '%s %s' % (data['class_number'], grh_lab)),
('Class group:', '%s %s' % (data['class_group_invs'], grh_lab)),
('Galois Group:', group_display_short(data['degree'], t, C))
]
from lmfdb.math_classes import NumberFieldGaloisGroup
try:
info["tim_number_field"] = NumberFieldGaloisGroup.find_one({"label": label})
v = nf.factor_perm_repn(info["tim_number_field"])
info["mydecomp"] = ['*' if x>0 else '' for x in v]
except AttributeError:
pass
# del info['_id']
return render_template("number_field.html", properties2=properties2, credit=NF_credit, title=title, bread=bread, friends=info.pop('friends'), learnmore=info.pop('learnmore'), info=info)
def render_hgm_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
C = base.getDBConnection()
data = C.hgm.motives.find_one({'label': label})
if data is None:
bread = get_bread([("Search error", url_for('.search'))])
info['err'] = "Motive " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
title = 'Hypergeometric Motive:' + label
A = data['A']
B = data['B']
tn, td = data['t']
t = latex(QQ(str(tn) + '/' + str(td)))
primes = [
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,
67, 71
]
locinfo = data['locinfo']
for j in range(len(locinfo)):
locinfo[j] = [primes[j]] + locinfo[j]
locinfo[j][2] = poly_with_factored_coeffs(locinfo[j][2], primes[j])
hodge = data['hodge']
prop2 = [
('Degree', '\(%s\)' % data['degree']),
('Weight', '\(%s\)' % data['weight']),
('Conductor', '\(%s\)' % data['cond']),
]
# Now add factorization of conductor
Cond = ZZ(data['cond'])
if not (Cond.abs().is_prime() or Cond == 1):
data['cond'] = "%s=%s" % (str(Cond), factorint(data['cond']))
info.update({
'A': A,
'B': B,
't': t,
'degree': data['degree'],
'weight': data['weight'],
'sign': data['sign'],
'sig': data['sig'],
'hodge': hodge,
'cond': data['cond'],
'req': data['req'],
'locinfo': locinfo
})
AB_data, t_data = data["label"].split("_t")
#AB_data = data["label"].split("_t")[0]
friends = [("Motive family " + AB_data.replace("_", " "),
url_for(".by_family_label", label=AB_data))]
friends.append(('L-function',
url_for("l_functions.l_function_hgm_page",
label=AB_data,
t='t' + t_data)))
# if rffriend != '':
# friends.append(('Discriminant root field', rffriend))
bread = get_bread([(label, ' ')])
return render_template("hgm-show-motive.html",
credit=HGM_credit,
title=title,
bread=bread,
info=info,
properties2=prop2,
friends=friends)
def galois_group_search(**args):
info = to_dict(args)
bread = get_bread([("Search results", url_for('.search'))])
C = base.getDBConnection()
query = {}
if 'jump_to' in info:
return render_group_webpage({'label': info['jump_to']})
for param in ['n', 't']:
if info.get(param):
info[param] = clean_input(info[param])
ran = info[param]
ran = ran.replace('..', '-')
if LIST_RE.match(ran):
tmp = parse_range2(ran, param)
else:
names = {'n': 'degree', 't': 't'}
info['err'] = 'Error parsing input for the %s. It needs to be an integer (such as 5), a range of integers (such as 2-10 or 2..10), or a comma-separated list of these (such as 2,3,8 or 3-5, 7, 8-11).' % names[param]
return search_input_error(info, bread)
# work around syntax for $or
# we have to foil out multiple or conditions
if tmp[0] == '$or' and '$or' in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query['$or']]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
query[tmp[0]] = tmp[1]
for param in ['cyc', 'solv', 'prim', 'parity']:
if info.get(param):
info[param] = str(info[param])
if info[param] == str(1):
query[param] = 1
elif info[param] == str(-1):
query[param] = -1 if param == 'parity' else 0
# Determine if we have any composite degrees
info['show_subs'] = True
if info.get('n'):
info['show_subs'] = False # now only show subs if a composite n is allowed
nparam = info.get('n')
nparam.replace('..', '-')
nlist = nparam.split(',')
found = False
for nl in nlist:
if '-' in nl:
inx = nl.index('-')
ll, hh = nl[:inx], nl[inx + 1:]
hh = int(hh)
jj = int(ll)
while jj <= hh and not found:
if not(ZZ(jj).is_prime() or ZZ(jj) == 1):
found = True
jj += 1
if found:
break
else:
jj = ZZ(nl)
if not (ZZ(jj).is_prime() or ZZ(jj) == 1):
found = True
break
if found:
info['show_subs'] = True
count_default = 50
if info.get('count'):
try:
count = int(info['count'])
except:
count = count_default
else:
count = count_default
info['count'] = count
start_default = 0
if info.get('start'):
try:
start = int(info['start'])
if(start < 0):
start += (1 - (start + 1) / count) * count
except:
start = start_default
else:
start = start_default
if info.get('paging'):
try:
paging = int(info['paging'])
if paging == 0:
start = 0
except:
pass
res = C.transitivegroups.groups.find(query).sort([('n', pymongo.ASCENDING), ('t', pymongo.ASCENDING)])
nres = res.count()
res = res.skip(start).limit(count)
if(start >= nres):
start -= (1 + (start - nres) / count) * count
if(start < 0):
start = 0
info['groups'] = res
info['group_display'] = group_display_prettyC(C)
info['report'] = "found %s groups" % nres
info['yesno'] = yesno
info['wgg'] = WebGaloisGroup.from_data
info['start'] = start
info['number'] = nres
if nres == 1:
info['report'] = 'unique match'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
return render_template("gg-search.html", info=info, title="Galois Group Search Result", bread=bread, credit=GG_credit)
def render_hgm_family_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
C = base.getDBConnection()
data = C.hgm.families.find_one({'label': label})
if data is None:
bread = get_bread([("Search error", url_for('.search'))])
info[
'err'] = "Family of hypergeometric motives " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
title = 'Hypergeometric Motive Family:' + label
A = data['A']
B = data['B']
hodge = data['hodge']
prop2 = [('Degree', '\(%s\)' % data['degree']),
('Weight', '\(%s\)' % data['weight'])]
info.update({
'A': A,
'B': B,
'degree': data['degree'],
'weight': data['weight'],
'hodge': hodge,
'gal2': data['gal2'],
'gal3': data['gal3'],
'gal5': data['gal5'],
'gal7': data['gal7'],
})
friends = [('Motives in the family',
url_for('hypergm.index') + "?A=%s&B=%s" % (str(A), str(B)))
]
# if unramfriend != '':
# friends.append(('Unramified subfield', unramfriend))
# if rffriend != '':
# friends.append(('Discriminant root field', rffriend))
info.update({
"plotcircle":
url_for(".hgm_family_circle_image",
AB="A" + ".".join(map(str, A)) + "_B" +
".".join(map(str, B)))
})
info.update({
"plotlinear":
url_for(".hgm_family_linear_image",
AB="A" + ".".join(map(str, A)) + "_B" +
".".join(map(str, B)))
})
info.update({
"plotconstant":
url_for(".hgm_family_constant_image",
AB="A" + ".".join(map(str, A)) + "_B" +
".".join(map(str, B)))
})
bread = get_bread([(label, ' ')])
return render_template("hgm-show-family.html",
credit=HGM_credit,
title=title,
bread=bread,
info=info,
properties2=prop2,
friends=friends)
def render_lattice_webpage(**args):
f = None
if 'label' in args:
lab = clean_input(args.get('label'))
if lab != args.get('label'):
return redirect(url_for('.render_lattice_webpage', label=lab), 301)
f = db.lat_lattices.lucky({'$or':[{'label': lab }, {'name': {'$contains': [lab]}}]})
if f is None:
t = "Integral Lattices Search Error"
bread = [('Lattices', url_for(".lattice_render_webpage"))]
flash_error("%s is not a valid label or name for an integral lattice in the database.", lab)
return render_template("lattice-error.html", title=t, properties=[], bread=bread, learnmore=learnmore_list())
info = {}
info.update(f)
info['friends'] = []
bread = [('Lattice', url_for(".lattice_render_webpage")), ('%s' % f['label'], ' ')]
credit = lattice_credit
info['dim']= int(f['dim'])
info['det']= int(f['det'])
info['level']=int(f['level'])
info['gram']=vect_to_matrix(f['gram'])
info['density']=str(f['density'])
info['hermite']=str(f['hermite'])
info['minimum']=int(f['minimum'])
info['kissing']=int(f['kissing'])
info['aut']=int(f['aut'])
if f['shortest']=="":
info['shortest']==f['shortest']
else:
if f['dim']==1:
info['shortest']=str(f['shortest']).strip('[').strip(']')
else:
if info['dim']*info['kissing']<100:
info['shortest']=[str([tuple(v)]).strip('[').strip(']').replace('),', '), ') for v in f['shortest']]
else:
max_vect_num=min(int(round(100/(info['dim']))), int(round(info['kissing']/2))-1);
info['shortest']=[str([tuple(f['shortest'][i])]).strip('[').strip(']').replace('),', '), ') for i in range(max_vect_num+1)]
info['all_shortest']="no"
info['download_shortest'] = [
(i, url_for(".render_lattice_webpage_download", label=info['label'], lang=i, obj='shortest_vectors')) for i in ['gp', 'magma','sage']]
if f['name']==['Leech']:
info['shortest']=[str([1,-2,-2,-2,2,-1,-1,3,3,0,0,2,2,-1,-1,-2,2,-2,-1,-1,0,0,-1,2]),
str([1,-2,-2,-2,2,-1,0,2,3,0,0,2,2,-1,-1,-2,2,-1,-1,-2,1,-1,-1,3]), str([1,-2,-2,-1,1,-1,-1,2,2,0,0,2,2,0,0,-2,2,-1,-1,-1,0,-1,-1,2])]
info['all_shortest']="no"
info['download_shortest'] = [
(i, url_for(".render_lattice_webpage_download", label=info['label'], lang=i, obj='shortest_vectors')) for i in ['gp', 'magma','sage']]
ncoeff=20
if f['theta_series'] != "":
coeff=[f['theta_series'][i] for i in range(ncoeff+1)]
info['theta_series']=my_latex(print_q_expansion(coeff))
info['theta_display'] = url_for(".theta_display", label=f['label'], number="")
info['class_number']=int(f['class_number'])
if f['dim']==1:
info['genus_reps']=str(f['genus_reps']).strip('[').strip(']')
else:
if info['dim']*info['class_number']<50:
info['genus_reps']=[vect_to_matrix(n) for n in f['genus_reps']]
else:
max_matrix_num=min(int(round(25/(info['dim']))), info['class_number']);
info['all_genus_rep']="no"
info['genus_reps']=[vect_to_matrix(f['genus_reps'][i]) for i in range(max_matrix_num+1)]
info['download_genus_reps'] = [
(i, url_for(".render_lattice_webpage_download", label=info['label'], lang=i, obj='genus_reps')) for i in ['gp', 'magma','sage']]
if f['name'] != "":
if f['name']==str(f['name']):
info['name']= str(f['name'])
else:
info['name']=str(", ".join(str(i) for i in f['name']))
else:
info['name'] == ""
info['comments']=str(f['comments'])
if 'Leech' in info['comments']: # no need to duplicate as it is in the name
info['comments'] = ''
if info['name'] == "":
t = "Integral Lattice %s" % info['label']
else:
t = "Integral Lattice "+info['label']+" ("+info['name']+")"
#This part code was for the dinamic knowl with comments, since the test is displayed this is redundant
# if info['name'] != "" or info['comments'] !="":
# info['knowl_args']= "name=%s&report=%s" %(info['name'], info['comments'].replace(' ', '-space-'))
info['properties'] = [
('Dimension', '%s' %info['dim']),
('Determinant', '%s' %info['det']),
('Level', '%s' %info['level'])]
if info['class_number'] == 0:
info['properties']=[('Class number', 'not available')]+info['properties']
else:
info['properties']=[('Class number', '%s' %info['class_number'])]+info['properties']
info['properties']=[('Label', '%s' % info['label'])]+info['properties']
if info['name'] != "" :
info['properties']=[('Name','%s' % info['name'] )]+info['properties']
# friends = [('L-series (not available)', ' ' ),('Half integral weight modular forms (not available)', ' ')]
return render_template("lattice-single.html", info=info, credit=credit, title=t, bread=bread, properties=info['properties'], learnmore=learnmore_list(), KNOWL_ID="lattice.%s"%info['label'])
def render_family(args):
info = {}
if 'label' in args:
label = clean_input(args['label'])
dataz = list(db.hgcwa_passports.search({'label': label}))
if not dataz:
flash_error("No family with label %s was found in the database.",
label)
return redirect(url_for(".index"))
data = dataz[0]
g = data['genus']
g0 = data['g0']
GG = ast.literal_eval(data['group'])
gn = GG[0]
gt = GG[1]
gp_string = str(gn) + '.' + str(gt)
pretty_group = sg_pretty(gp_string)
if gp_string == pretty_group:
spname = False
else:
spname = True
title = 'Family of Genus ' + str(
g) + ' Curves with Automorphism Group $' + pretty_group + '$'
smallgroup = "[" + str(gn) + "," + str(gt) + "]"
prop2 = [
('Genus', r'\(%d\)' % g), ('Quotient Genus', r'\(%d\)' % g0),
('Group', r'\(%s\)' % pretty_group),
('Signature',
r'\(%s\)' % sign_display(ast.literal_eval(data['signature'])))
]
info.update({
'genus': data['genus'],
'sign': sign_display(ast.literal_eval(data['signature'])),
'group': pretty_group,
'g0': data['g0'],
'dim': data['dim'],
'r': data['r'],
'gpid': smallgroup
})
if spname:
info.update({'specialname': True})
Lcc = []
Lall = []
i = 1
for dat in dataz:
if ast.literal_eval(dat['con']) not in Lcc:
urlstrng = dat['passport_label']
Lcc.append(ast.literal_eval(dat['con']))
Lall.append([
cc_display(ast.literal_eval(dat['con'])),
dat['passport_label'], urlstrng
])
i = i + 1
info.update({'passport': Lall})
g2List = [
'[2,1]', '[4,2]', '[8,3]', '[10,2]', '[12,4]', '[24,8]', '[48,29]'
]
if g == 2 and data['group'] in g2List:
g2url = "/Genus2Curve/Q/?geom_aut_grp_id=" + data['group']
friends = [(r"Genus 2 curves over $\Q$", g2url)]
else:
friends = []
br_g, br_gp, br_sign = split_family_label(label)
bread_sign = label_to_breadcrumbs(br_sign)
bread_gp = label_to_breadcrumbs(br_gp)
bread = get_bread([(br_g, './?genus=' + br_g),
('$' + pretty_group + '$',
'./?genus=' + br_g + '&group=' + bread_gp),
(bread_sign, ' ')])
learnmore = [('Completeness of the data',
url_for(".completeness_page")),
('Source of the data', url_for(".how_computed_page")),
('Reliability of the data', url_for(".reliability_page")),
('Labeling convention', url_for(".labels_page"))]
downloads = [('Code to Magma',
url_for(".hgcwa_code_download",
label=label,
download_type='magma')),
('Code to Gap',
url_for(".hgcwa_code_download",
label=label,
download_type='gap'))]
return render_template("hgcwa-show-family.html",
title=title,
bread=bread,
info=info,
properties=prop2,
friends=friends,
learnmore=learnmore,
downloads=downloads,
credit=credit)
def render_group_webpage(args):
data = {}
if 'label' in args:
label = clean_input(args['label'])
label = label.replace('t', 'T')
data = db.gps_transitive.lookup(label)
if data is None:
if re.match(r'^\d+T\d+$', label):
flash_error("Group %s was not found in the database.", label)
else:
flash_error("%s is not a valid label for a Galois group.",
label)
return redirect(url_for(".index"))
title = 'Galois group: ' + label
wgg = WebGaloisGroup.from_nt(data['n'], data['t'])
data['wgg'] = wgg
n = data['n']
t = data['t']
data['yesno'] = yesno
order = data['order']
data['orderfac'] = latex(ZZ(order).factor())
orderfac = latex(ZZ(order).factor())
data['ordermsg'] = "$%s=%s$" % (order, latex(orderfac))
if order == 1:
data['ordermsg'] = "$1$"
if ZZ(order).is_prime():
data['ordermsg'] = "$%s$ (is prime)" % order
pgroup = len(ZZ(order).prime_factors()) < 2
if wgg.num_conjclasses() < 50:
data['cclasses'] = wgg.conjclasses()
if ZZ(order) < ZZ(10000000) and wgg.num_conjclasses() < 21:
data['chartable'] = chartable(n, t)
data['gens'] = wgg.generator_string()
if n == 1 and t == 1:
data['gens'] = 'None needed'
data['num_cc'] = comma(wgg.num_conjclasses())
data['parity'] = "$%s$" % data['parity']
data['subinfo'] = subfield_display(n, data['subfields'])
data['resolve'] = resolve_display(data['quotients'])
if data['gapid'] == 0:
data['gapid'] = "not available"
else:
data['gapid'] = small_group_display_knowl(
int(data['order']), int(data['gapid']),
str([int(data['order']),
int(data['gapid'])]))
data['otherreps'] = wgg.otherrep_list()
ae = data['arith_equiv']
if ae > 0:
if ae > 1:
data[
'arith_equiv'] = r'A number field with this Galois group has %d <a knowl="nf.arithmetically_equivalent", title="arithmetically equivalent">arithmetically equivalent</a> fields.' % ae
else:
data[
'arith_equiv'] = r'A number field with this Galois group has exactly one <a knowl="nf.arithmetically_equivalent", title="arithmetically equivalent">arithmetically equivalent</a> field.'
elif ae > -1:
data[
'arith_equiv'] = r'A number field with this Galois group has no <a knowl="nf.arithmetically_equivalent", title="arithmetically equivalent">arithmetically equivalent</a> fields.'
else:
data[
'arith_equiv'] = r'Data on whether or not a number field with this Galois group has <a knowl="nf.arithmetically_equivalent", title="arithmetically equivalent">arithmetically equivalent</a> fields has not been computed.'
intreps = list(db.gps_gmodules.search({'n': n, 't': t}))
if len(intreps) > 0:
data['int_rep_classes'] = [str(z[0]) for z in intreps[0]['gens']]
for onerep in intreps:
onerep['gens'] = [
list_to_latex_matrix(z[1]) for z in onerep['gens']
]
data['int_reps'] = intreps
data['int_reps_complete'] = int_reps_are_complete(intreps)
dcq = data['moddecompuniq']
if dcq[0] == 0:
data['decompunique'] = 0
else:
data['decompunique'] = dcq[0]
data['isoms'] = [[mult2mult(z[0]),
mult2mult(z[1])] for z in dcq[1]]
data['isoms'] = [[
modules2string(n, t, z[0]),
modules2string(n, t, z[1])
] for z in data['isoms']]
#print dcq[1]
#print data['isoms']
friends = []
if db.nf_fields.exists({'galois_label': "%dT%d" % (n, t)}):
friends.append(
('Number fields with this Galois group',
url_for('number_fields.number_field_render_webpage') +
"?galois_group=%dT%d" % (n, t)))
prop2 = [
('Label', label),
('Degree', prop_int_pretty(data['n'])),
('Order', prop_int_pretty(order)),
('Cyclic', yesno(data['cyc'])),
('Abelian', yesno(data['ab'])),
('Solvable', yesno(data['solv'])),
('Primitive', yesno(data['prim'])),
('$p$-group', yesno(pgroup)),
]
pretty = group_display_short(n, t, emptyifnotpretty=True)
if len(pretty) > 0:
prop2.extend([('Group:', pretty)])
data['pretty_name'] = pretty
data['name'] = re.sub(r'_(\d+)', r'_{\1}', data['name'])
data['name'] = re.sub(r'\^(\d+)', r'^{\1}', data['name'])
data['nilpotency'] = '$%s$' % data['nilpotency']
if data['nilpotency'] == '$-1$':
data['nilpotency'] += ' (not nilpotent)'
bread = get_bread([(label, ' ')])
return render_template("gg-show-group.html",
credit=GG_credit,
title=title,
bread=bread,
info=data,
properties=prop2,
friends=friends,
KNOWL_ID="gg.%s" % label,
learnmore=learnmore_list())
def render_passport(args):
info = {}
if 'passport_label' in args:
label = clean_input(args['passport_label'])
dataz = list(db.hgcwa_passports.search({'passport_label': label}))
if not dataz:
bread = get_bread([("Search Error", url_for('.index'))])
flash_error(
"No refined passport with label %s was found in the database.",
label)
return redirect(url_for(".index"))
data = dataz[0]
g = data['genus']
g0 = data['g0']
GG = ast.literal_eval(data['group'])
gn = GG[0]
gt = GG[1]
gp_string = str(gn) + '.' + str(gt)
pretty_group = sg_pretty(gp_string)
if gp_string == pretty_group:
spname = False
else:
spname = True
numb = len(dataz)
try:
numgenvecs = request.args['numgenvecs']
numgenvecs = int(numgenvecs)
except:
numgenvecs = 20
info['numgenvecs'] = numgenvecs
title = 'One Refined Passport of Genus ' + str(
g) + ' with Automorphism Group $' + pretty_group + '$'
smallgroup = "[" + str(gn) + "," + str(gt) + "]"
prop2 = [
('Genus', r'\(%d\)' % g), ('Quotient Genus', r'\(%d\)' % g0),
('Group', r'\(%s\)' % pretty_group),
('Signature',
r'\(%s\)' % sign_display(ast.literal_eval(data['signature']))),
('Generating Vectors', r'\(%d\)' % numb)
]
info.update({
'genus': data['genus'],
'cc': cc_display(data['con']),
'sign': sign_display(ast.literal_eval(data['signature'])),
'group': pretty_group,
'gpid': smallgroup,
'numb': numb,
'disp_numb': min(numb, numgenvecs),
'g0': data['g0']
})
if spname:
info.update({'specialname': True})
Ldata = []
HypColumn = False
Lfriends = []
for i in range(0, min(numgenvecs, numb)):
dat = dataz[i]
x1 = dat['total_label']
if 'full_auto' in dat:
x2 = 'No'
if dat['full_label'] not in Lfriends:
Lfriends.append(dat['full_label'])
else:
x2 = 'Yes'
if 'hyperelliptic' in dat:
x3 = tfTOyn(dat['hyperelliptic'])
HypColumn = True
else:
x3 = ' '
x4 = []
if dat['g0'] == 0:
for perm in dat['gen_vectors']:
cycperm = Permutation(perm).cycle_string()
x4.append(sep.join(split_perm(cycperm)))
elif dat['g0'] > 0:
for perm in dat['gen_vectors']:
cycperm = Permutation(perm).cycle_string()
#if display_perm == '()':
if cycperm == '()':
x4.append('Id(G)')
else:
x4.append(sep.join(split_perm(cycperm)))
Ldata.append([x1, x2, x3, x4])
info.update({'genvects': Ldata, 'HypColumn': HypColumn})
info.update({'passport_cc': cc_display(ast.literal_eval(data['con']))})
if 'eqn' in data:
info.update({'eqns': data['eqn']})
if 'ndim' in data:
info.update({'Ndim': data['ndim']})
other_data = False
if 'hyperelliptic' in data:
info.update({'ishyp': tfTOyn(data['hyperelliptic'])})
other_data = True
if 'hyp_involution' in data:
inv = Permutation(data['hyp_involution']).cycle_string()
info.update({'hypinv': sep.join(split_perm(inv))})
if 'cyclic_trigonal' in data:
info.update({'iscyctrig': tfTOyn(data['cyclic_trigonal'])})
other_data = True
if 'jacobian_decomp' in data:
jcLatex, corrChar = decjac_format(data['jacobian_decomp'])
info.update({'corrChar': corrChar, 'jacobian_decomp': jcLatex})
if 'cinv' in data:
cinv = Permutation(data['cinv']).cycle_string()
info.update({'cinv': sep.join(split_perm(cinv))})
info.update({'other_data': other_data})
if 'full_auto' in data:
full_G = ast.literal_eval(data['full_auto'])
full_gn = full_G[0]
full_gt = full_G[1]
full_gp_string = str(full_gn) + '.' + str(full_gt)
full_pretty_group = sg_pretty(full_gp_string)
info.update({
'fullauto': full_pretty_group,
'signH': sign_display(ast.literal_eval(data['signH'])),
'higgenlabel': data['full_label']
})
urlstrng, br_g, br_gp, br_sign, _ = split_passport_label(label)
if Lfriends:
friends = [("Full automorphism " + Lf, Lf) for Lf in Lfriends]
friends += [("Family containing this refined passport ", urlstrng)]
else:
friends = [("Family containing this refined passport", urlstrng)]
bread_sign = label_to_breadcrumbs(br_sign)
bread_gp = label_to_breadcrumbs(br_gp)
bread = get_bread([(br_g, './?genus=' + br_g),
('$' + pretty_group + '$',
'./?genus=' + br_g + '&group=' + bread_gp),
(bread_sign, urlstrng), (data['cc'][0], ' ')])
learnmore = [('Completeness of the data',
url_for(".completeness_page")),
('Source of the data', url_for(".how_computed_page")),
('Reliability of the data', url_for(".reliability_page")),
('Labeling convention', url_for(".labels_page"))]
downloads = [('Code to Magma',
url_for(".hgcwa_code_download",
label=label,
download_type='magma')),
('Code to Gap',
url_for(".hgcwa_code_download",
label=label,
download_type='gap'))]
return render_template("hgcwa-show-passport.html",
title=title,
bread=bread,
info=info,
properties=prop2,
friends=friends,
learnmore=learnmore,
downloads=downloads,
credit=credit)
def render_group_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
label = label.replace('t', 'T')
C = base.getDBConnection()
data = C.transitivegroups.groups.find_one({'label': label})
if data is None:
bread = get_bread([("Search error", url_for('.search'))])
info['err'] = "Group " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
title = 'Galois Group:' + label
n = data['n']
t = data['t']
data['yesno'] = yesno
order = data['order']
data['orderfac'] = latex(ZZ(order).factor())
orderfac = latex(ZZ(order).factor())
data['ordermsg'] = "$%s=%s$" % (order, latex(orderfac))
if ZZ(order) == 1:
data['ordermsg'] = "$1$"
if ZZ(order).is_prime():
data['ordermsg'] = "$%s$ (is prime)" % order
pgroup = len(ZZ(order).prime_factors()) < 2
if n == 1:
G = gap.SmallGroup(n, t)
else:
G = gap.TransitiveGroup(n, t)
if ZZ(order) < ZZ('10000000000'):
ctable = chartable(n, t)
else:
ctable = 'Group too large'
data['gens'] = generators(n, t)
if n == 1 and t == 1:
data['gens'] = 'None needed'
data['chartable'] = ctable
data['parity'] = "$%s$" % data['parity']
data['cclasses'] = conjclasses(G, n)
data['subinfo'] = subfield_display(C, n, data['subs'])
data['resolve'] = resolve_display(C, data['resolve'])
# if len(data['resolve']) == 0: data['resolve'] = 'None'
data['otherreps'] = otherrep_display(n, t, C, data['repns'])
prop2 = [
('Order:', '\(%s\)' % order),
('n:', '\(%s\)' % data['n']),
('Cyclic:', yesno(data['cyc'])),
('Abelian:', yesno(data['ab'])),
('Solvable:', yesno(data['solv'])),
('Primitive:', yesno(data['prim'])),
('$p$-group:', yesno(pgroup)),
('Name:', group_display_short(n, t, C)),
]
info.update(data)
bread = get_bread([(label, ' ')])
return render_template("gg-show-group.html",
credit=GG_credit,
title=title,
bread=bread,
info=info,
properties2=prop2)
def render_field_webpage(args):
data = None
C = base.getDBConnection()
info = {}
bread = [('Global Number Fields', url_for(".number_field_render_webpage"))]
if 'label' in args:
label = clean_input(args['label'])
nf = WebNumberField(label)
data = {}
if nf.is_null():
bread.append(('Search results', ' '))
label2 = re.sub(r'[<>]', '', args['label'])
if 'You need to enter a field' in label2:
info['err'] = label2
else:
info['err'] = 'No such field: %s in the database' % label2
info['label'] = args['label_orig'] if 'label_orig' in args else args[
'label']
return search_input_error(info, bread)
info['wnf'] = nf
from lmfdb.WebNumberField import nf_display_knowl
data['degree'] = nf.degree()
data['class_number'] = nf.class_number()
t = nf.galois_t()
n = nf.degree()
data['is_galois'] = nf.is_galois()
data['is_abelian'] = nf.is_abelian()
if nf.is_abelian():
conductor = nf.conductor()
data['conductor'] = conductor
dirichlet_chars = nf.dirichlet_group()
if len(dirichlet_chars) > 0:
data['dirichlet_group'] = [
'<a href = "%s">$\chi_{%s}(%s,·)$</a>' %
(url_for('characters.render_Dirichletwebpage',
modulus=data['conductor'],
number=j), data['conductor'], j)
for j in dirichlet_chars
]
data['dirichlet_group'] = r'$\lbrace$' + ', '.join(
data['dirichlet_group']) + r'$\rbrace$'
if data['conductor'].is_prime() or data['conductor'] == 1:
data['conductor'] = "\(%s\)" % str(data['conductor'])
else:
data['conductor'] = "\(%s=%s\)" % (str(
data['conductor']), latex(data['conductor'].factor()))
data['galois_group'] = group_display_knowl(n, t, C)
data['cclasses'] = cclasses_display_knowl(n, t, C)
data['character_table'] = character_table_display_knowl(n, t, C)
data['class_group'] = nf.class_group()
data['class_group_invs'] = nf.class_group_invariants()
data['signature'] = nf.signature()
data['coefficients'] = nf.coeffs()
D = nf.disc()
ram_primes = D.prime_factors()
data['disc_factor'] = nf.disc_factored_latex()
if D.abs().is_prime() or D == 1:
data['discriminant'] = "\(%s\)" % str(D)
else:
data['discriminant'] = "\(%s=%s\)" % (str(D), data['disc_factor'])
npr = len(ram_primes)
ram_primes = str(ram_primes)[1:-1]
if ram_primes == '':
ram_primes = r'\textrm{None}'
data['frob_data'], data['seeram'] = frobs(nf.K())
data['phrase'] = group_phrase(n, t, C)
zk = nf.zk()
Ra = PolynomialRing(QQ, 'a')
zk = [latex(Ra(x)) for x in zk]
zk = ['$%s$' % x for x in zk]
zk = ', '.join(zk)
grh_label = '<small>(<a title="assuming GRH" knowl="nf.assuming_grh">assuming GRH</a>)</small>' if nf.used_grh(
) else ''
# Short version for properties
grh_lab = nf.short_grh_string()
if 'Not' in str(data['class_number']):
grh_lab = ''
grh_label = ''
pretty_label = field_pretty(label)
if label != pretty_label:
pretty_label = "%s: %s" % (label, pretty_label)
info.update(data)
info.update({
'label':
pretty_label,
'label_raw':
label,
'polynomial':
web_latex_split_on_pm(nf.K().defining_polynomial()),
'ram_primes':
ram_primes,
'integral_basis':
zk,
'regulator':
web_latex(nf.regulator()),
'unit_rank':
nf.unit_rank(),
'root_of_unity':
web_latex(nf.K().primitive_root_of_unity()),
'fund_units':
nf.units(),
'grh_label':
grh_label
})
bread.append(('%s' % info['label_raw'], ' '))
info['downloads_visible'] = True
info['downloads'] = [('worksheet', '/')]
info['friends'] = []
if nf.can_class_number():
info['friends'].append(('L-function', "/L/NumberField/%s" % label))
info['friends'].append(('Galois group', "/GaloisGroup/%dT%d" % (n, t)))
if 'dirichlet_group' in info:
info['friends'].append(('Dirichlet group',
url_for("characters.dirichlet_group_table",
modulus=int(conductor),
char_number_list=','.join(
[str(a) for a in dirichlet_chars]),
poly=info['polynomial'])))
info['learnmore'] = [('Global number field labels',
url_for(".render_labels_page")),
(Completename, url_for(".render_discriminants_page")),
('Quadratic imaginary class groups',
url_for(".render_class_group_data"))]
# With Galois group labels, probably not needed here
# info['learnmore'] = [('Global number field labels',
# url_for(".render_labels_page")), ('Galois group
# labels',url_for(".render_groups_page")),
# (Completename,url_for(".render_discriminants_page"))]
title = "Global Number Field %s" % info['label']
if npr == 1:
primes = 'prime'
else:
primes = 'primes'
properties2 = [
('Degree:', '%s' % data['degree']),
('Signature:', '$%s$' % data['signature']),
('Discriminant:', '$%s$' % data['disc_factor']),
('Ramified ' + primes + ':', '$%s$' % ram_primes),
('Class number:', '%s %s' % (data['class_number'], grh_lab)),
('Class group:', '%s %s' % (data['class_group_invs'], grh_lab)),
('Galois Group:', group_display_short(data['degree'], t, C))
]
from lmfdb.math_classes import NumberFieldGaloisGroup
try:
info["tim_number_field"] = NumberFieldGaloisGroup.find_one(
{"label": label})
v = nf.factor_perm_repn(info["tim_number_field"])
info["mydecomp"] = ['*' if x > 0 else '' for x in v]
except AttributeError:
pass
# del info['_id']
return render_template("number_field.html",
properties2=properties2,
credit=NF_credit,
title=title,
bread=bread,
friends=info.pop('friends'),
learnmore=info.pop('learnmore'),
info=info)
def elliptic_curve_search(**args):
info = to_dict(args)
query = {}
bread = [('Elliptic Curves', url_for(".rational_elliptic_curves")),
('Search Results', '.')]
if 'jump' in args:
label = info.get('label', '').replace(" ", "")
m = lmfdb_label_regex.match(label)
if m:
try:
return by_ec_label(label)
except ValueError:
return elliptic_curve_jump_error(label,
info,
wellformed_label=True)
elif label.startswith("Cremona:"):
label = label[8:]
m = cremona_label_regex.match(label)
if m:
try:
return by_ec_label(label)
except ValueError:
return elliptic_curve_jump_error(label,
info,
wellformed_label=True)
elif cremona_label_regex.match(label):
return elliptic_curve_jump_error(label, info, cremona_label=True)
elif label:
# Try to parse a string like [1,0,3,2,4]
lab = re.sub(r'\s', '', label)
lab = re.sub(r'^\[', '', lab)
lab = re.sub(r']$', '', lab)
try:
labvec = lab.split(',')
labvec = [QQ(str(z)) for z in labvec] # Rationals allowed
E = EllipticCurve(labvec)
ainvs = [str(c) for c in E.minimal_model().ainvs()]
C = lmfdb.base.getDBConnection()
data = C.elliptic_curves.curves.find_one({'ainvs': ainvs})
if data is None:
return elliptic_curve_jump_error(label, info)
return by_ec_label(data['lmfdb_label'])
except (ValueError, ArithmeticError):
return elliptic_curve_jump_error(label, info)
else:
query['label'] = ''
if info.get('jinv'):
j = clean_input(info['jinv'])
j = j.replace('+', '')
if not QQ_RE.match(j):
info[
'err'] = 'Error parsing input for the j-invariant. It needs to be a rational number.'
return search_input_error(info, bread)
query['jinv'] = j
for field in ['conductor', 'torsion', 'rank', 'sha_an']:
if info.get(field):
info[field] = clean_input(info[field])
ran = info[field]
ran = ran.replace('..', '-').replace(' ', '')
if not LIST_RE.match(ran):
names = {
'conductor': 'conductor',
'torsion': 'torsion order',
'rank': 'rank',
'sha_an': 'analytic order of Ш'
}
info[
'err'] = 'Error parsing input for the %s. It needs to be an integer (such as 5), a range of integers (such as 2-10 or 2..10), or a comma-separated list of these (such as 2,3,8 or 3-5, 7, 8-11).' % names[
field]
return search_input_error(info, bread)
# Past input check
tmp = parse_range2(ran, field)
# work around syntax for $or
# we have to foil out multiple or conditions
if tmp[0] == '$or' and '$or' in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query['$or']]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
if field == 'sha_an': # database sha_an values are not all exact!
query[tmp[0]] = {'$gt': tmp[1] - 0.1, '$lt': tmp[1] + 0.1}
else:
query[tmp[0]] = tmp[1]
if 'optimal' in info and info['optimal'] == 'on':
# fails on 990h3
query['number'] = 1
if 'torsion_structure' in info and info['torsion_structure']:
info['torsion_structure'] = clean_input(info['torsion_structure'])
if not TORS_RE.match(info['torsion_structure']):
info[
'err'] = 'Error parsing input for the torsion structure. It needs to be one or more integers in square brackets, such as [6], [2,2], or [2,4]. Moreover, each integer should be bigger than 1, and each divides the next.'
return search_input_error(info, bread)
query['torsion_structure'] = [
str(a) for a in parse_list(info['torsion_structure'])
]
if info.get('surj_primes'):
info['surj_primes'] = clean_input(info['surj_primes'])
format_ok = LIST_POSINT_RE.match(info['surj_primes'])
if format_ok:
surj_primes = [int(p) for p in info['surj_primes'].split(',')]
format_ok = all([ZZ(p).is_prime(proof=False) for p in surj_primes])
if format_ok:
query['non-surjective_primes'] = {"$nin": surj_primes}
else:
info[
'err'] = 'Error parsing input for surjective primes. It needs to be a prime (such as 5), or a comma-separated list of primes (such as 2,3,11).'
return search_input_error(info, bread)
if info.get('nonsurj_primes'):
info['nonsurj_primes'] = clean_input(info['nonsurj_primes'])
format_ok = LIST_POSINT_RE.match(info['nonsurj_primes'])
if format_ok:
nonsurj_primes = [
int(p) for p in info['nonsurj_primes'].split(',')
]
format_ok = all(
[ZZ(p).is_prime(proof=False) for p in nonsurj_primes])
if format_ok:
if info['surj_quantifier'] == 'exactly':
nonsurj_primes.sort()
query['non-surjective_primes'] = nonsurj_primes
else:
if 'non-surjective_primes' in query:
query['non-surjective_primes'] = {
"$nin": surj_primes,
"$all": nonsurj_primes
}
else:
query['non-surjective_primes'] = {"$all": nonsurj_primes}
else:
info[
'err'] = 'Error parsing input for nonsurjective primes. It needs to be a prime (such as 5), or a comma-separated list of primes (such as 2,3,11).'
return search_input_error(info, bread)
info['query'] = query
count_default = 100
if info.get('count'):
try:
count = int(info['count'])
except:
count = count_default
else:
count = count_default
info['count'] = count
start_default = 0
if info.get('start'):
try:
start = int(info['start'])
if (start < 0):
start += (1 - (start + 1) / count) * count
except:
start = start_default
else:
start = start_default
cursor = lmfdb.base.getDBConnection().elliptic_curves.curves.find(query)
nres = cursor.count()
if (start >= nres):
start -= (1 + (start - nres) / count) * count
if (start < 0):
start = 0
res = cursor.sort([('conductor', ASCENDING), ('lmfdb_iso', ASCENDING),
('lmfdb_number', ASCENDING)]).skip(start).limit(count)
info['curves'] = res
info['format_ainvs'] = format_ainvs
info['number'] = nres
info['start'] = start
if nres == 1:
info['report'] = 'unique match'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (
start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
credit = 'John Cremona'
if 'non-surjective_primes' in query:
credit += 'and Andrew Sutherland'
t = 'Elliptic Curves'
return render_template("search_results.html",
info=info,
credit=credit,
bread=bread,
title=t)
def number_field_search(**args):
info = to_dict(args)
info['learnmore'] = [
('Global number field labels', url_for(".render_labels_page")),
('Galois group labels', url_for(".render_groups_page")),
(Completename, url_for(".render_discriminants_page")),
('Quadratic imaginary class groups',
url_for(".render_class_group_data"))
]
t = 'Global Number Field search results'
bread = [('Global Number Fields', url_for(".number_field_render_webpage")),
('Search results', ' ')]
# for k in info.keys():
# nf_logger.debug(str(k) + ' ---> ' + str(info[k]))
# nf_logger.debug('******************* '+ str(info['search']))
if 'natural' in info:
field_id = info['natural']
field_id_parsed = parse_field_string(info['natural'])
if FIELD_LABEL_RE.match(field_id_parsed):
field_id_parsed = split_label(
field_id_parsed) # allows factored labels 11.11.11e20.1
return render_field_webpage({
'label': field_id_parsed,
'label_orig': field_id
})
query = {}
dlist = []
for field in [
'galois_group', 'degree', 'signature', 'discriminant',
'class_number', 'class_group'
]:
if info.get(field):
info[field] = clean_input(info[field])
if field in ['class_group', 'signature']:
# different regex for the two types
if (field == 'signature' and PAIR_RE.match(info[field])) or (
field == 'class_group' and IF_RE.match(info[field])):
query[field] = info[field][1:-1]
else:
name = 'class group' if field == 'class_group' else 'signature'
info[
'err'] = 'Error parsing input for %s. It needs to be a pair of integers in square brackets, such as [2,3] or [3,3]' % name
return search_input_error(info, bread)
else:
if field == 'galois_group':
try:
gcs = complete_group_codes(info[field])
if len(gcs) == 1:
query['galois'] = make_galois_pair(
gcs[0][0], gcs[0][1])
# list(gcs[0])
if len(gcs) > 1:
query['galois'] = {
'$in':
[make_galois_pair(x[0], x[1]) for x in gcs]
}
except NameError as code:
info[
'err'] = 'Error parsing input for Galois group: unknown group label %s. It needs to be a <a title = "Galois group labels" knowl="nf.galois_group.name">group label</a>, such as C5 or 5T1, or comma separated list of labels.' % code
return search_input_error(info, bread)
else: # not signature, class group, or galois group
ran = info[field]
ran = ran.replace('..', '-')
if LIST_RE.match(ran):
if field == 'discriminant':
# dlist will contain the disc conditions
# as sage ints
dlist = parse_discs(ran)
# now convert to a query
tmp = list_to_query(dlist)
# Two cases, could be a list of sign/inequalties
# or an '$or'
if len(tmp) == 1:
tmp = tmp[0]
else:
query[tmp[0][0]] = tmp[0][1]
tmp = tmp[1]
else:
tmp = parse_range2(ran, field)
# work around syntax for $or
# we have to foil out multiple or conditions
if tmp[0] == '$or' and '$or' in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query['$or']]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
query[tmp[0]] = tmp[1]
else:
name = re.sub('_', ' ', field)
info[
'err'] = 'Error parsing input for %s. It needs to be an integer (such as 5), a range of integers (such as 2-100 or 2..100), or a comma-separated list of these (such as 2,3,8 or 3-5, 7, 8-100).' % name
return search_input_error(info, bread)
if info.get('ur_primes'):
# now we want a list of strings, no spaces, which might be big ints
info['ur_primes'] = clean_input(info['ur_primes'])
if LIST_SIMPLE_RE.match(info['ur_primes']):
ur_primes = info['ur_primes'].split(',')
# Assuming this will be the only nor in the query
query['$nor'] = [{'ramps': x} for x in ur_primes]
else:
info[
'err'] = 'Error parsing input for unramified primes. It needs to be an integer (such as 5), or a comma-separated list of integers (such as 2,3,11).'
return search_input_error(info, bread)
if info.get('ram_primes'):
# now we want a list of strings, no spaces, which might be big ints
info['ram_primes'] = clean_input(info['ram_primes'])
if LIST_SIMPLE_RE.match(info['ram_primes']):
ram_primes = info['ram_primes'].split(',')
if str(info['ram_quantifier']) == 'some':
query['ramps'] = {'$all': ram_primes}
else:
query['ramps'] = ram_primes
else:
info[
'err'] = 'Error parsing input for ramified primes. It needs to be an integer (such as 5), or a comma-separated list of integers (such as 2,3,11).'
return search_input_error(info, bread)
count_default = 20
if info.get('count'):
try:
count = int(info['count'])
except:
count = count_default
info['count'] = count
else:
info['count'] = count_default
count = count_default
info['count'] = int(info['count'])
start_default = 0
if info.get('start'):
try:
start = int(info['start'])
if (start < 0):
start += (1 - (start + 1) / count) * count
except:
start = start_default
else:
start = start_default
if info.get('paging'):
try:
paging = int(info['paging'])
if paging == 0:
start = 0
except:
pass
C = base.getDBConnection()
# nf_logger.debug(query)
info['query'] = dict(query)
if 'lucky' in args:
one = C.numberfields.fields.find_one(query)
if one:
label = one['label']
return render_field_webpage({'label': label})
fields = C.numberfields.fields
res = fields.find(query).sort([('degree', ASC), ('disc_abs_key', ASC),
('disc_sign', ASC), ('label', ASC)])
if 'download' in info and info['download'] != '0':
return download_search(info, res)
nres = res.count()
res = res.skip(start).limit(count)
if (start >= nres):
start -= (1 + (start - nres) / count) * count
if (start < 0):
start = 0
info['fields'] = res
info['number'] = nres
info['start'] = start
if nres == 1:
info['report'] = 'unique match'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (
start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
info['wnf'] = WebNumberField.from_data
return render_template("number_field_search.html",
info=info,
title=t,
bread=bread)
def render_hecke_algebras_webpage_l_adic(**args):
data = None
if 'orbit_label' in args and 'prime' in args:
lab = clean_input(args.get('orbit_label'))
if lab != args.get('orbit_label'):
base_lab = ".".join([split(lab)[i] for i in [0, 1, 2]])
return redirect(
url_for('.render_hecke_algebras_webpage', label=base_lab), 301)
try:
ell = int(args.get('prime'))
except ValueError:
base_lab = ".".join([split(lab)[i] for i in [0, 1, 2]])
return redirect(
url_for('.render_hecke_algebras_webpage', label=base_lab), 301)
data = db.hecke_ladic.lucky({'orbit_label': lab, 'ell': ell})
if data is None:
t = "Hecke algebra search error"
bread = [('HeckeAlgebra', url_for(".hecke_algebras_render_webpage"))]
flash_error(
"%s is not a valid label for the ℓ-adic information for an Hecke algebra orbit in the database.",
lab)
return render_template("hecke_algebras-error.html",
title=t,
properties=[],
bread=bread,
learnmore=learnmore_list())
info = {}
info.update(data)
proj = [
'index', 'orbit_label', 'ell', 'idempotent', 'field', 'structure',
'properties', 'operators'
]
res = list(
db.hecke_ladic.search(
{
'level': data['level'],
'weight': data['weight'],
'orbit_label': data['orbit_label'],
'ell': data['ell']
}, proj))
for f in res:
if f['idempotent'] != "":
dim = len(sage_eval(f['idempotent']))
l_max = sage_eval(f['idempotent'])[0][0].ndigits()
if dim > 4 or l_max > 5:
f['idempotent_display'] = []
elif dim == 1:
f['idempotent_display'] = sage_eval(f['idempotent'])[0][0]
else:
f['idempotent_display'] = latex(
matrix(sage_eval(f['idempotent'])))
else:
f['idempotent_display'] = latex(matrix([[1]]))
del f['idempotent']
f['download_id'] = [
(i,
url_for(".render_hecke_algebras_webpage_ell_download",
orbit_label=f['orbit_label'],
index=f['index'],
prime=f['ell'],
lang=i,
obj='idempotents')) for i in ['magma', 'sage']
] # for 'gp' the code does not work, since p-adics are not implemented
field = f.pop('field')
if field is not None:
f['deg'] = field[1]
f['field_poly'] = field[2]
structure = f.pop('structure')
if structure is not None:
f['dim'] = structure[0]
f['num_gen'] = structure[1]
s2 = sage_eval(structure[2])
f['gens'] = [[int(s2.index(i) + 1), str(i)] for i in s2]
f['rel'] = sage_eval(structure[3])
properties = f.pop('properties')
if properties is not None:
f['grading'] = properties[0]
f['gorenstein_def'] = properties[1]
f['gorenstein'] = "yes" if properties[1] == 0 else "no"
operators = f.pop('operators')
if operators is not None:
f['operators_mod_l'] = operators
f['num_hecke_op'] = len(operators)
f['size_op'] = size = sqrt(len(operators[0]))
if size > 4:
f['operators_mod_l_display'] = []
elif size == 1:
f['operators_mod_l_display'] = [[i + 1, operators[i][0]]
for i in range(10)]
else:
f['operators_mod_l_display'] = [[
i + 1, latex(matrix(size, size, operators[i]))
] for i in range(5)]
f['download_op'] = [
(lang,
url_for(".render_hecke_algebras_webpage_ell_download",
orbit_label=f['orbit_label'],
index=f['index'],
prime=f['ell'],
lang=lang,
obj='operators')) for lang in ['magma', 'sage']
] # for 'gp' the code does not work
info['num_l_adic_orbits'] = len(res)
info['l_adic_orbits'] = res
info['level'] = int(data['level'])
info['weight'] = int(data['weight'])
info['base_lab'] = ".".join(
[split(data['orbit_label'])[i] for i in [0, 1, 2]])
info['orbit_label'] = str(data['orbit_label'])
info['ell'] = int(data['ell'])
bread = [('HeckeAlgebra', url_for(".hecke_algebras_render_webpage")),
('%s' % info['base_lab'],
url_for('.render_hecke_algebras_webpage',
label=info['base_lab'])), ('%s' % info['ell'], ' ')]
credit = hecke_algebras_credit
info['properties'] = [('Level', '%s' % info['level']),
('Weight', '%s' % info['weight']),
('Characteristic', '%s' % info['ell']),
('Orbit label', '%s' % info['orbit_label'])]
info['friends'] = [('Modular form ' + info['base_lab'],
url_for("cmf.by_url_space_label",
level=info['level'],
weight=info['weight'],
char_orbit_label='a'))]
t = "%s-adic and mod %s data for the Hecke algebra orbit %s" % (
info['ell'], info['ell'], info['orbit_label'])
return render_template("hecke_algebras_l_adic-single.html",
info=info,
credit=credit,
title=t,
bread=bread,
properties=info['properties'],
learnmore=learnmore_list(),
friends=info['friends'],
KNOWL_ID='hecke_algebra_l_adic.%s' %
(info['orbit_label']))
def elliptic_curve_search(**args):
info = to_dict(args['data'])
if 'jump' in info:
label = info.get('label', '').replace(" ", "")
# This label should be a full isogeny class label or a full
# curve label (including the field_label component)
try:
nf, cond_label, iso_label, number = split_full_label(label.strip())
except IndexError:
if not 'query' in info:
info['query'] = {}
bread = [("Elliptic Curves", url_for(".index"))]
info[
'err'] = 'No elliptic curve in the database has label %s.' % label
return search_input_error(info, bread)
return show_ecnf(nf, cond_label, iso_label, number)
query = {}
bread = [('Elliptic Curves', url_for(".index")), ('Search Results', '.')]
if 'conductor_norm' in info:
Nnorm = clean_input(info['conductor_norm'])
Nnorm = Nnorm.replace('..', '-').replace(' ', '')
tmp = parse_range2(Nnorm, 'conductor_norm')
if tmp[0] == '$or' and '$or' in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query['$or']]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
query[tmp[0]] = tmp[1]
if 'conductor_label' in info:
query['conductor_label'] = info['conductor_label']
if 'jinv' in info:
query['jinv'] = info['jinv']
if info.get('torsion'):
ran = info['torsion'] = clean_input(info['torsion'])
ran = ran.replace('..', '-').replace(' ', '')
if not LIST_RE.match(ran):
info[
'err'] = 'Error parsing input for the torsion order. It needs to be an integer (such as 5), a range of integers (such as 2-10 or 2..10), or a comma-separated list of these (such as 4,9,16 or 4-25, 81-121).'
return search_input_error(info, bread)
# Past input check
tmp = parse_range2(ran, 'torsion_order')
# work around syntax for $or
# we have to foil out multiple or conditions
if tmp[0] == '$or' and '$or' in query:
newors = []
for y in tmp[1]:
oldors = [dict.copy(x) for x in query['$or']]
for x in oldors:
x.update(y)
newors.extend(oldors)
tmp[1] = newors
query[tmp[0]] = tmp[1]
if 'torsion_structure' in info and info['torsion_structure']:
res = parse_torsion_structure(info['torsion_structure'], 2)
if 'Error' in res:
info['err'] = res
return search_input_error(info, bread)
#update info for repeat searches
info['torsion_structure'] = str(res).replace(' ', '')
query['torsion_structure'] = [int(r) for r in res]
if 'include_isogenous' in info and info['include_isogenous'] == 'off':
query['number'] = 1
if 'include_base_change' in info and info['include_base_change'] == 'off':
query['base_change'] = []
else:
info['include_base_change'] = "on"
if 'field' in info:
query['field_label'] = parse_field_string(info['field'])
info['query'] = query
# process count and start if not default:
count_default = 50
if info.get('count'):
try:
count = int(info['count'])
except:
count = count_default
else:
count = count_default
start_default = 0
if info.get('start'):
try:
start = int(info['start'])
if (start < 0):
start += (1 - (start + 1) / count) * count
except:
start = start_default
else:
start = start_default
# make the query and trim results according to start/count:
cursor = db_ecnf().find(query)
nres = cursor.count()
if (start >= nres):
start -= (1 + (start - nres) / count) * count
if (start < 0):
start = 0
res = cursor.sort([('field_label', ASC), ('conductor_norm', ASC),
('conductor_label', ASC), ('iso_nlabel', ASC),
('number', ASC)]).skip(start).limit(count)
res = list(res)
for e in res:
e['numb'] = str(e['number'])
e['field_knowl'] = nf_display_knowl(e['field_label'],
getDBConnection(),
e['field_label'])
info['curves'] = res # [ECNF(e) for e in res]
info['number'] = nres
info['start'] = start
info['count'] = count
info['more'] = int(start + count < nres)
info['field_pretty'] = field_pretty
info['web_ainvs'] = web_ainvs
if nres == 1:
info['report'] = 'unique match'
else:
if nres > count or start != 0:
info['report'] = 'displaying matches %s-%s of %s' % (
start + 1, min(nres, start + count), nres)
else:
info['report'] = 'displaying all %s matches' % nres
t = 'Elliptic Curve search results'
return render_template("ecnf-search-results.html",
info=info,
credit=ecnf_credit,
bread=bread,
title=t)
