def galois_string(self):
if not self.haskey('galois'):
return 'Not computed'
n = self._data['degree']
t = self._data['galois']['t']
C = base.getDBConnection()
return group_display_short(n, t, C)
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 galois_string(self):
if not self.haskey('galt'):
return 'Not computed'
n = self._data['degree']
t = self._data['galt']
return group_display_short(n, t)
def render_field_webpage(args):
data = None
info = {}
bread = [('Global Number Fields', url_for(".number_field_render_webpage"))]
# This function should not be called unless label is set.
label = clean_input(args['label'])
nf = WebNumberField(label)
data = {}
if nf.is_null():
bread.append(('Search Results', ' '))
info['err'] = 'There is no field with label %s in the database' % label
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_latex()
ram_primes = nf.ramified_primes()
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:
factored_conductor = factor_base_factor(data['conductor'], ram_primes)
factored_conductor = factor_base_factorization_latex(factored_conductor)
data['conductor'] = "\(%s=%s\)" % (str(data['conductor']), factored_conductor)
data['galois_group'] = group_display_knowl(n, t)
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'] = "\(%s\)" % str(D)
else:
data['discriminant'] = "\(%s=%s\)" % (str(D), data['disc_factor'])
data['frob_data'], data['seeram'] = frobs(nf)
# 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])
else:
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]
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]
if 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 '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)
if nf.degree() > 1:
gpK = nf.gpK()
rootof1coeff = gpK.nfrootsof1()
rootofunityorder = int(rootof1coeff[1])
rootof1coeff = rootof1coeff[2]
rootofunity = web_latex(Ra(str(pari("lift(%s)" % gpK.nfbasistoalg(rootof1coeff))).replace('x','a')))
rootofunity += ' (order $%d$)' % rootofunityorder
else:
rootofunity = web_latex(Ra('-1'))+ ' (order $2$)'
info.update({
'label': pretty_label,
'label_raw': label,
'polynomial': web_latex_split_on_pm(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(),
'grh_label': grh_label,
'loc_alg': loc_alg
})
bread.append(('%s' % 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 len(galois_closure[1])>0:
resinfo.append(('gc', galois_closure[1]))
if len(galois_closure[2]) > 0:
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 len(sextic_twins[1])>0:
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 len(siblings[0])>0:
for sibdeg in siblings[0]:
if len(sibdeg[2]) ==0:
sibdeg[2] = dnc
else:
sibdeg[2] = ', '.join(sibdeg[2])
if len(sibdeg[2])<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 len(arith_equiv[1])>0:
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()
#if info['signature'] == [0,1]:
# info['learnmore'].append(('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 = [('Label', label),
('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))
]
downloads = []
for lang in [["Magma","magma"], ["SageMath","sage"], ["Pari/GP", "gp"]]:
downloads.append(('Download {} code'.format(lang[0]),
url_for(".nf_code_download", nf=label, download_type=lang[1])))
from lmfdb.artin_representations.math_classes import NumberFieldGaloisGroup
try:
info["tim_number_field"] = NumberFieldGaloisGroup(nf._data['coeffs'])
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("number_field.html", properties2=properties2, credit=NF_credit, title=title, bread=bread, code=nf.code, friends=info.pop('friends'), downloads=downloads, learnmore=learnmore, info=info)
def group(self):
return group_display_short(self._data['Galn'],self._data['Galt'])
def gds(nt):
return group_display_short(nt[0], nt[1], C)
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 gds(nt):
return group_display_short(nt['n'], nt['t'], C)
def gds(nt):
return group_display_short(nt[0], nt[1], C)
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", ' ')])
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 = [
('Label', label),
('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': {
'$in': [1, 2]
},
'rf': data['rf']
})
if rfdata is None:
logger.fatal("Cannot find discriminant root field!")
rffriend = ''
else:
rffriend = "/LocalNumberField/%s" % rfdata['label']
info.update({
'polynomial': web_latex(polynomial),
'n': data['n'],
'p': data['p'],
'c': data['c'],
'e': data['e'],
'f': data['f'],
't': data['t'],
'u': data['u'],
'rf': printquad(data['rf'], p),
'hw': data['hw'],
'slopes': show_slopes(data['slopes']),
'gal': group_display_knowl(gn, gt, C),
'gt': gt,
'inertia': group_display_inertia(data['inertia'], C),
'unram': web_latex(unramp),
'eisen': web_latex(eisenp),
'gms': data['gms'],
'aut': data['aut'],
})
friends = [('Galois group', "/GaloisGroup/%dT%d" % (gn, gt))]
if unramfriend != '':
friends.append(('Unramified subfield', unramfriend))
if rffriend != '':
friends.append(('Discriminant root field', rffriend))
bread = get_bread([(label, ' ')])
learnmore = [('Completeness of the data',
url_for(".completeness_page")),
('Source of the data', url_for(".how_computed_page")),
('Local field labels', url_for(".labels_page"))]
return render_template("lf-show-field.html",
credit=LF_credit,
title=title,
bread=bread,
info=info,
properties2=prop2,
friends=friends,
learnmore=learnmore)
def render_group_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
label = label.replace('t', 'T')
C = base.getDBConnection()
data = C.transitivegroups.groups.find_one({'label': label})
if data is None:
bread = get_bread([("Search error", url_for('.search'))])
info['err'] = "Group " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
title = 'Galois Group:' + label
wgg = WebGaloisGroup.from_data(data)
n = data['n']
t = data['t']
data['yesno'] = yesno
order = data['order']
data['orderfac'] = latex(ZZ(order).factor())
orderfac = latex(ZZ(order).factor())
data['ordermsg'] = "$%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']
]
#onerep.update({'gens': onerep['gens'][1]})
#onerep.update({'gens': [str(onerep['gens'][0]), list_to_latex_matrix(onerep['gens'][1]])})
data['int_reps'] = 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 render_field_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
data = lfdb().find_one({'label': label})
if data is None:
bread = get_bread([("Search error", ' ')])
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(string2list(data['coeffs']))
p = data['p']
e = data['e']
f = data['f']
cc = data['c']
GG = data['gal']
gn = GG[0]
gt = GG[1]
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', '\(\Q_{%s}\)' % p),
('Degree', '\(%s\)' % data['n']),
('e', '\(%s\)' % e),
('f', '\(%s\)' % f),
('c', '\(%s\)' % cc),
('Galois group', group_display_short(gn, gt, db())),
]
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 = lfdb().find_one({'p': p, 'n': f, 'c': 0})
if unramdata is not None:
unramfriend = "/LocalNumberField/%s" % unramdata['label']
else:
logger.fatal("Cannot find unramified field!")
unramfriend = ''
rfdata = lfdb().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']
gsm = data['gsm']
if gsm == '0':
gsm = 'Not computed'
elif gsm == '-1':
gsm = 'Does not exist'
else:
gsm = web_latex(coeff_to_poly(string2list(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': printquad(data['rf'], p),
'hw': data['hw'],
'slopes': show_slopes(data['slopes']),
'gal': group_display_knowl(gn, gt, db()),
'gt': gt,
'inertia': group_display_inertia(data['inertia'], db()),
'unram': web_latex(unramp),
'eisen': web_latex(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, ' ')])
learnmore = [('Completeness of the data', url_for(".completeness_page")),
('Source of the data', url_for(".how_computed_page")),
('Local field labels', url_for(".labels_page"))]
return render_template("lf-show-field.html", credit=LF_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends, learnmore=learnmore)
def gds(nt):
return group_display_short(nt['n'], nt['t'], C)
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:
bread = get_bread([("Search Error", ' ')])
info['err'] = "Field " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
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_display_short(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_display_knowl(gn, gt),
'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, ' ')])
learnmore = [('Completeness of the data', url_for(".completeness_page")),
('Source of the data', url_for(".how_computed_page")),
('Local field labels', url_for(".labels_page"))]
return render_template("lf-show-field.html", credit=LF_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends, learnmore=learnmore)
def galois_string(self):
n = self._data["degree"]
t = self._data["galois"]["t"]
C = base.getDBConnection()
return group_display_short(n, t, C)
def galois_string(self):
n = self._data['degree']
t = self._data['galois']['t']
C = base.getDBConnection()
return group_display_short(n, t, C)
def group(self):
return group_display_short(self._data["Galois_nt"][0], self._data["Galois_nt"][1], getDBConnection())
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)),
('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 render_field_webpage(args):
data = None
C = base.getDBConnection()
info = {}
bread = [('Global Number Fields', url_for(".number_field_render_webpage"))]
# This function should not be called unless label is set.
label = clean_input(args['label'])
nf = WebNumberField(label)
data = {}
if nf.is_null():
bread.append(('Search results', ' '))
info['err'] = 'There is no field with label %s in the database' % label
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_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()
nf.make_code_snippets()
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)
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)
if nf.degree() > 1:
gpK = nf.gpK()
rootof1coeff = gpK.nfrootsof1()[2]
rootofunity = Ra(str(pari("lift(%s)" % gpK.nfbasistoalg(rootof1coeff))).replace('x','a'))
else:
rootofunity = Ra('-1')
info.update({
'label': pretty_label,
'label_raw': label,
'polynomial': web_latex_split_on_pm(nf.poly()),
'ram_primes': ram_primes,
'integral_basis': zk,
'regulator': web_latex(nf.regulator()),
'unit_rank': nf.unit_rank(),
'root_of_unity': web_latex(rootofunity),
'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():
# 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 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")),
('How data was computed', url_for(".how_computed_page"))]
if info['signature'] == [0,1]:
info['learnmore'].append(('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 = [('Label', label),
('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(nf._data['coeffs'])
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
# del info['_id']
return render_template("number_field.html", properties2=properties2, credit=NF_credit, title=title, bread=bread, code=nf.code, friends=info.pop('friends'), learnmore=info.pop('learnmore'), info=info)
def group(self):
return group_display_short(self._data['Galois_nt'][0],
self._data['Galois_nt'][1],
getDBConnection())
def render_field_webpage(args):
data = None
info = {}
bread = [('Global Number Fields', url_for(".number_field_render_webpage"))]
# This function should not be called unless label is set.
label = clean_input(args['label'])
nf = WebNumberField(label)
data = {}
if nf.is_null():
bread.append(('Search Results', ' '))
info['err'] = 'There is no field with label %s in the database' % label
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_latex()
ram_primes = nf.ramified_primes()
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:
factored_conductor = factor_base_factor(data['conductor'],
ram_primes)
factored_conductor = factor_base_factorization_latex(
factored_conductor)
data['conductor'] = "\(%s=%s\)" % (str(
data['conductor']), factored_conductor)
data['galois_group'] = group_display_knowl(n, t)
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'] = "\(%s\)" % str(D)
else:
data['discriminant'] = "\(%s=%s\)" % (str(D), data['disc_factor'])
data['frob_data'], data['seeram'] = frobs(nf)
# 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])
else:
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]
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]
if 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 '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)
if nf.degree() > 1:
gpK = nf.gpK()
rootof1coeff = gpK.nfrootsof1()
rootofunityorder = int(rootof1coeff[1])
rootof1coeff = rootof1coeff[2]
rootofunity = web_latex(
Ra(
str(pari("lift(%s)" % gpK.nfbasistoalg(rootof1coeff))).replace(
'x', 'a')))
rootofunity += ' (order $%d$)' % rootofunityorder
else:
rootofunity = web_latex(Ra('-1')) + ' (order $2$)'
info.update({
'label': pretty_label,
'label_raw': label,
'polynomial': web_latex_split_on_pm(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(),
'grh_label': grh_label,
'loc_alg': loc_alg
})
bread.append(('%s' % 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 len(galois_closure[1]) > 0:
resinfo.append(('gc', galois_closure[1]))
if len(galois_closure[2]) > 0:
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 len(sextic_twins[1]) > 0:
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 len(siblings[0]) > 0:
for sibdeg in siblings[0]:
if len(sibdeg[2]) == 0:
sibdeg[2] = dnc
else:
sibdeg[2] = ', '.join(sibdeg[2])
if len(sibdeg[2]) < 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 len(arith_equiv[1]) > 0:
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()
#if info['signature'] == [0,1]:
# info['learnmore'].append(('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 = [('Label', label), ('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))]
downloads = []
for lang in [["Magma", "magma"], ["SageMath", "sage"], ["Pari/GP", "gp"]]:
downloads.append(('Download {} code'.format(lang[0]),
url_for(".nf_code_download",
nf=label,
download_type=lang[1])))
from lmfdb.artin_representations.math_classes import NumberFieldGaloisGroup
try:
info["tim_number_field"] = NumberFieldGaloisGroup(nf._data['coeffs'])
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("number_field.html",
properties2=properties2,
credit=NF_credit,
title=title,
bread=bread,
code=nf.code,
friends=info.pop('friends'),
downloads=downloads,
learnmore=learnmore,
info=info)
def group(self):
return group_display_short(self._data['Galn'], self._data['Galt'])
def render_field_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
data = lfdb().find_one({'label': label})
if data is None:
bread = get_bread([("Search error", ' ')])
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(string2list(data['coeffs']))
p = data['p']
e = data['e']
f = data['f']
cc = data['c']
GG = data['gal']
gn = GG[0]
gt = GG[1]
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', '\(\Q_{%s}\)' % p),
('Degree', '\(%s\)' % data['n']),
('e', '\(%s\)' % e),
('f', '\(%s\)' % f),
('c', '\(%s\)' % cc),
('Galois group', group_display_short(gn, gt, db())),
]
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 = lfdb().find_one({'p': p, 'n': f, 'c': 0})
if unramdata is not None:
unramfriend = "/LocalNumberField/%s" % unramdata['label']
else:
logger.fatal("Cannot find unramified field!")
unramfriend = ''
rfdata = lfdb().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']
gsm = data['gsm']
if gsm == '0':
gsm = 'Not computed'
elif gsm == '-1':
gsm = 'Does not exist'
else:
gsm = web_latex(coeff_to_poly(string2list(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': printquad(data['rf'], p),
'hw': data['hw'],
'slopes': show_slopes(data['slopes']),
'gal': group_display_knowl(gn, gt, db()),
'gt': gt,
'inertia': group_display_inertia(data['inertia'], db()),
'unram': web_latex(unramp),
'eisen': web_latex(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, ' ')])
learnmore = [('Completeness of the data', url_for(".completeness_page")),
('Source of the data', url_for(".how_computed_page")),
('Local field labels', url_for(".labels_page"))]
return render_template("lf-show-field.html", credit=LF_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends, learnmore=learnmore)
def galois_string(self):
if not self.haskey('galt'):
return 'Not computed'
n = self._data['degree']
t = self._data['galt']
return group_display_short(n, t)
def galois_string(self):
n = self._data['degree']
t = self._data['galois']['t']
C = base.getDBConnection()
return group_display_short(n, t, C)
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", " ")])
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 = [
("Label", label),
("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, " ")])
learnmore = [
("Completeness of the data", url_for(".completeness_page")),
("Source of the data", url_for(".how_computed_page")),
("Local field labels", url_for(".labels_page")),
]
return render_template(
"lf-show-field.html",
credit=LF_credit,
title=title,
bread=bread,
info=info,
properties2=prop2,
friends=friends,
learnmore=learnmore,
)