def parse_artin_label(label):
label = clean_input(label)
if re.compile(r'^\d+\.\d+e\d+(_\d+e\d+)*\.\d+t\d\.\d+c\d+$').match(label):
return label
else:
raise ValueError(
"Error parsing input %s. It is not in a valid form for an Artin representation label, such as 9.2e12_587e3.10t32.1c1"
% label)
def parse_artin_label(label):
label = clean_input(label)
if LABEL_RE.match(label):
return label
else:
raise ValueError(
"Error parsing input %s. It is not in a valid form for an Artin representation label, such as 9.2e12_587e3.10t32.1c1"
% label)
def hgm_jump(info):
label = clean_input(info['jump_to'])
if HGM_LABEL_RE.match(label):
return render_hgm_webpage(normalize_motive(label))
if HGM_FAMILY_LABEL_RE.match(label):
return render_hgm_family_webpage(normalize_family(label))
flash_error('%s is not a valid label for a hypergeometric motive or family of hypergeometric motives', label)
return redirect(url_for(".index"))
def parse_artin_label(label):
label = clean_input(label)
if re.compile(r"^\d+\.\d+e\d+(_\d+e\d+)*\.\d+t\d\.\d+c\d+$").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 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 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': nf_string_to_label(label)})
except ValueError as err:
flash(Markup("Error: <span style='color:black'>%s</span> is not a valid number field. %s" % (label,err)), "error")
bread = [('Global Number Fields', url_for(".number_field_render_webpage")), ('Search Results', ' ')]
return search_input_error({'err':''}, bread)
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': nf_string_to_label(label)})
except ValueError as err:
flash(Markup("Error: <span style='color:black'>%s</span> is not a valid number field. %s" % (label,err)), "error")
bread = [('Global Number Fields', url_for(".number_field_render_webpage")), ('Search Results', ' ')]
return search_input_error({'err':''}, bread)
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_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'])
def render_hgcwa_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
C = base.getDBConnection()
data = C.curve_automorphisms.families.find_one({'label': label})
if data is None:
bread = get_bread([("Search error", url_for('.search'))])
info[
'err'] = "Higher Genus Curve with Automorphism " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
g = data['genus']
GG = data['trangplabel']
gn = GG[0]
gt = GG[1]
group = groupid_to_meaningful(data['group'])
title = 'Family of genus ' + str(
g) + ' curves with automorphism group $' + group + '$'
sign = data['signature']
prop2 = [('Genus', '\(%d\)' % g),
('Group', group_display_short(gn, gt, C)),
('Signature', '\(%s\)' % sign)]
info.update({
'genus': data['genus'],
'genvecs': perm_display(data['gen_vectors']),
'sign': data['signature'],
'group': groupid_to_meaningful(data['group']),
'g0': data['g0'],
'dim': data['r'] - 3,
'r': data['r'],
'ishyp': tfTOyn(data['hyperelliptic']),
'deg': data['degree'],
'trgroup': group_display_short(gn, gt, C)
})
if 'hyp_eqn' in data:
info.update({'eqn': data['hyp_eqn']})
gg = "/GaloisGroup/"
if 'full_auto' in data:
info.update({
'fullauto': groupid_to_meaningful(data['full_auto']),
'signH': data['signH'],
'higgenlabel': data['full_label']
})
higgenstrg = "/HigherGenus/C/aut/" + data['full_label']
friends = [('Automorphism group', gg),
('Family of full automorphism', higgenstrg)]
else:
friends = [('Automorphism group', gg)]
bread = get_bread([(label, ' ')])
learnmore = [('Completeness of the data',
url_for(".completeness_page")),
('Source of the data', url_for(".how_computed_page")),
('Labeling convension', url_for(".labels_page"))]
downloads = [('Download this example', '.')]
return render_template("hgcwa-show-curve.html",
credit=HGCwA_credit,
title=title,
bread=bread,
info=info,
properties2=prop2,
friends=friends,
learnmore=learnmore,
downloads=downloads)
def render_artin_representation_webpage(label):
if re.compile(r'^\d+$').match(label):
return artin_representation_search(**{'dimension': label})
bread = get_bread([(label, ' ')])
# label=dim.cond.nTt.indexcj, c is literal, j is index in conj class
# Should we have a big try around this to catch bad labels?
label = clean_input(label)
try:
the_rep = ArtinRepresentation(label)
except:
flash(
Markup(
"Error: <span style='color:black'>%s</span> is not the label of an Artin representation in the database."
% (label)), "error")
return search_input_error({'err': ''}, bread)
extra_data = {} # for testing?
C = getDBConnection()
extra_data['galois_knowl'] = group_display_knowl(5, 3, C) # for testing?
#artin_logger.info("Found %s" % (the_rep._data))
title = "Artin representation %s" % label
the_nf = the_rep.number_field_galois_group()
from lmfdb.number_field_galois_groups import nfgg_page
from lmfdb.number_field_galois_groups.main import by_data
if the_rep.sign() == 0:
processed_root_number = "not computed"
else:
processed_root_number = str(the_rep.sign())
properties = [
("Dimension", str(the_rep.dimension())),
("Group", the_rep.group()),
#("Conductor", str(the_rep.conductor())),
("Conductor", "$" + 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 cc.modulus <= 100000:
if the_rep.dimension() == 1:
friends.append(("Corresponding Dirichlet character",
url_for("characters.render_Dirichletwebpage",
modulus=cc.modulus,
number=cc.number)))
else:
friends.append(("Determinant character",
url_for("characters.render_Dirichletwebpage",
modulus=cc.modulus,
number=cc.number)))
# once the L-functions are in the database, the link can always be shown
if the_rep.dimension() <= 6:
friends.append(("L-function",
url_for("l_functions.l_function_artin_page",
label=the_rep.label())))
info = {}
#mychar = the_rep.central_char()
#info['pol2']= str([((j+1),mychar(j+1, 2*the_rep.character_field())) for j in range(50)])
#info['pol3']=str(the_rep.central_character())
#info['pol3']=str(the_rep.central_char(3))
#info['pol5']=str(the_rep.central_char(5))
#info['pol7']=str(the_rep.central_char(7))
#info['pol11']=str(the_rep.central_char(11))
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)
def render_family(args):
info = {}
if 'label' in args:
label = clean_input(args['label'])
C = base.getDBConnection()
dataz = C.curve_automorphisms.passports.find({'label': label}).sort('cc.0', pymongo.ASCENDING)
if dataz.count() is 0:
flash_error( "No family with label %s was found in the database.", label)
return redirect(url_for(".index"))
data=dataz[0]
g = data['genus']
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', '\(%d\)' % g),
('Group', '\(%s\)' % pretty_group),
('Signature', '\(%s\)' % 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=[]
Ltopo_rep=[] #List of topological representatives
Lelements=[] #List of lists of equivalence classes
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])
#Topological equivalence
if 'topological' in dat:
if dat['topological'] == dat['cc']:
x1=[] #A list of permutations of generating vectors of topo_rep
for perm in dat['gen_vectors']:
x1.append(sep.join(split_perm(Permutation(perm).cycle_string())))
Ltopo_rep.append([dat['passport_label'], dat['total_label'], x1])
topo_class = C.curve_automorphisms.passports.find({'label': dat['label'], 'topological': dat['cc']}).sort('cc.0', pymongo.ASCENDING)
elements=[] #An equivalence class
for element in topo_class:
elements.append((element['passport_label'], element['total_label']))
Lelements.append(elements)
Ltopo_class = zip(Ltopo_rep, Lelements)
topo_length = len(Ltopo_rep)
#Add topological equivalence to info
info.update({'topological_rep': Ltopo_rep})
info.update({'topological_class': Ltopo_class})
info.update({'topological_num': topo_length})
info.update({'passport': Lall})
info.update({'passport_num': len(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 = [("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")),
('Labeling convention', url_for(".labels_page"))]
if topo_length == 0:
downloads = [('Download Magma code', url_for(".hgcwa_code_download", label=label, download_type='magma')),
('Download Gap code', url_for(".hgcwa_code_download", label=label, download_type='gap'))]
else:
downloads = [('Magma code', None),
(u'\u2003 All vectors', url_for(".hgcwa_code_download", label=label, download_type='magma')),
(u'\u2003 Up to topological equivalence', url_for(".hgcwa_code_download", label=label, download_type='topo_magma')),
('Gap code', None),
(u'\u2003 All vectors', url_for(".hgcwa_code_download", label=label, download_type='gap')),
(u'\u2003 Up to topological equivalence', url_for(".hgcwa_code_download", label=label, download_type='topo_gap'))]
return render_template("hgcwa-show-family.html",
title=title, bread=bread, info=info,
properties2=prop2, friends=friends,
learnmore=learnmore, downloads=downloads, credit=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 = hecke_db().find_one({'label': lab })
if data is None:
t = "Hecke Agebras 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 a Hecke Algebras in the database." % (lab)),"error")
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
f = hecke_db().find_one({'level': data['level'],'weight': data['weight'],'num_orbits': data['num_orbits']})
info['level']=int(f['level'])
info['weight']= int(f['weight'])
info['num_orbits']= int(f['num_orbits'])
dim_count = "not available"
orb = hecke_orbits_db().find({'parent_label': f['label']}).sort([('orbit', ASC)])
if orb.count()!=0:
#consistency check
if orb.count()!= int(f['num_orbits']):
return search_input_error(info)
dim_count=0
res_clean = []
for v in orb:
v_clean = {}
v_clean['orbit_label'] = v['orbit_label']
v_clean['hecke_op'] = v['hecke_op']
v_clean['gen'] = v['gen']
v_clean['dim'] = int(matrix(sage_eval(v_clean['hecke_op'])[0]).dimensions()[0])
dim_count = dim_count + v_clean['dim']
if v_clean['dim']>4:
v_clean['hecke_op_display'] = []
elif v_clean['dim']==1:
v_clean['hecke_op_display'] = [[i+1, (sage_eval(v_clean['hecke_op'])[i])[0][0]] for i in range(0,10)]
else:
v_clean['hecke_op_display'] = [[i+1, latex(matrix(sage_eval(v_clean['hecke_op'])[i]))] for i in range(0,5)]
v_clean['num_hecke_op'] = v['num_hecke_op']
v_clean['download_op'] = [(i, url_for(".render_hecke_algebras_webpage_download", orbit_label=v_clean['orbit_label'], lang=i, obj='operators')) for i in ['gp', 'magma','sage']]
if 'Zbasis' in v.keys():
v_clean['Zbasis'] = [[int(i) for i in j] for j in v['Zbasis']]
if v_clean['dim']>4:
v_clean['gen_display'] = []
elif v_clean['dim']==1:
v_clean['gen_display'] = [v_clean['Zbasis'][0][0]]
else:
v_clean['gen_display'] = [latex(matrix(v_clean['dim'],v_clean['dim'], v_clean['Zbasis'][i])) for i in range(0,v_clean['dim'])]
v_clean['discriminant'] = int(v['discriminant'])
v_clean['disc_fac'] = [[int(i) for i in j] for j in v['disc_fac']]
v_clean['Qbasis'] = [int(i) for i in v['Qbasis']]
v_clean['Qalg_gen'] = [int(i) for i in v['Qalg_gen']]
if 'inner_twists' in v.keys():
v_clean['inner_twists'] = [str(i) for i in v['inner_twists']]
else:
v_clean['inner_twists'] = "not available"
v_clean['download_gen'] = [(i, url_for(".render_hecke_algebras_webpage_download", orbit_label=v_clean['orbit_label'], lang=i, obj='gen')) for i in ['gp', 'magma','sage']]
res_clean.append(v_clean)
info['orbits'] = res_clean
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("emf.render_elliptic_modular_forms", level=info['level'], weight=info['weight'], character=1))]
else:
info['friends'] = []
t = "Hecke Algebra %s" % info['label']
return render_template("hecke_algebras-single.html", info=info, credit=credit, title=t, bread=bread, properties2=info['properties'], learnmore=learnmore_list(), friends=info['friends'])
def render_family(args):
info = {}
if 'label' in args:
label = clean_input(args['label'])
C = base.getDBConnection()
dataz = C.curve_automorphisms.passports.find({'label': label})
if dataz.count() is 0:
flash_error("No family with label %s was found in the database.",
label)
return redirect(url_for(".index"))
data = dataz[0]
g = data['genus']
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', '\(%d\)' % g), ('Group', '\(%s\)' % pretty_group),
('Signature',
'\(%s\)' % 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 = [("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")),
('Labeling convention', url_for(".labels_page"))]
downloads = [('Download Magma code',
url_for(".hgcwa_code_download",
label=label,
download_type='magma')),
('Download Gap code',
url_for(".hgcwa_code_download",
label=label,
download_type='gap'))]
return render_template("hgcwa-show-family.html",
title=title,
bread=bread,
info=info,
properties2=prop2,
friends=friends,
learnmore=learnmore,
downloads=downloads,
credit=credit)
def number_field_search(info):
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', ' ')]
if 'natural' in info:
query = {'label_orig': info['natural']}
try:
parse_nf_string(info,
query,
'natural',
name="Label",
qfield='label')
return redirect(
url_for(".by_label", label=clean_input(query['label_orig'])))
except ValueError:
query['err'] = info['err']
return search_input_error(query, bread)
if 'algebra' in info:
fields = info['algebra'].split('_')
fields2 = [WebNumberField.from_coeffs(a) for a in fields]
for j in range(len(fields)):
if fields2[j] is None:
fields2[j] = WebNumberField.fakenf(fields[j])
t = 'Number field algebra'
info = {}
info = {'fields': fields2}
return render_template("number_field_algebra.html",
info=info,
title=t,
bread=bread)
query = {}
try:
parse_galgrp(info, query, qfield='galois')
parse_ints(info, query, 'degree')
parse_bracketed_posints(info,
query,
'signature',
split=False,
exactlength=2)
parse_signed_ints(info,
query,
'discriminant',
qfield=('disc_sign', 'disc_abs_key'),
parse_one=make_disc_key)
parse_ints(info, query, 'class_number')
parse_bracketed_posints(info,
query,
'class_group',
split=False,
check_divisibility='increasing')
parse_primes(info,
query,
'ur_primes',
name='Unramified primes',
qfield='ramps',
mode='complement',
to_string=True)
# modes are now contained (in), exactly, include
if 'ram_quantifier' in info and str(
info['ram_quantifier']) == 'include':
mode = 'append'
parse_primes(info,
query,
'ram_primes',
'ramified primes',
'ramps',
mode,
to_string=True)
elif 'ram_quantifier' in info and str(
info['ram_quantifier']) == 'contained':
parse_primes(info,
query,
'ram_primes',
'ramified primes',
'ramps_all',
'subsets',
to_string=False)
pass # build list
else:
mode = 'liststring'
parse_primes(info, query, 'ram_primes', 'ramified primes',
'ramps_all', mode)
except ValueError:
return search_input_error(info, bread)
count = parse_count(info)
start = parse_start(info)
# nf_logger.debug(query)
info['query'] = dict(query)
if 'lucky' in info:
one = nfdb().find_one(query)
if one:
label = one['label']
return redirect(url_for(".by_label", label=clean_input(label)))
fields = nfdb()
res = fields.find(query)
res = res.sort([('degree', ASC), ('disc_abs_key', ASC),
('disc_sign', 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 by_label(label):
clean_label = clean_input(label)
if clean_label != label:
return redirect(url_for('.by_label', label=clean_label), 301)
return search_by_label(label)
def render_hgcwa_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
C = base.getDBConnection()
data = C.curve_automorphisms.families.find_one({'label': label})
if data is None:
bread = get_bread([("Search error", url_for('.search'))])
info['err'] = "Higher Genus Curve with Automorphism " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
g = data['genus']
GG = data['group']
gn = GG[0]
gt = GG[1]
group = groupid_to_meaningful(data['group'])
if group == str(GG) or group == "[" + str(gn)+","+str(gt)+"]":
spname=False
else:
spname=True
title = 'Family of genus ' + str(g) + ' curves with automorphism group $' + group +'$'
smallgroup="(" + str(gn) + "," +str(gt) +")"
prop2 = [
('Genus', '\(%d\)' % g),
('Small Group', '\(%s\)' % smallgroup),
('Signature', '\(%s\)' % sign_display(data['signature']))
]
info.update({'genus': data['genus'],
'genvecs': data['gen_vectors'],
'sign': sign_display(data['signature']),
'group': groupid_to_meaningful(data['group']),
'g0':data['g0'],
'dim':data['dim'],
'r':data['r'],
'gpid': smallgroup
})
if spname:
info.update({'specialname': True})
if 'eqn' in data:
info.update({'eqn': data['eqn']})
if 'hyperelliptic' in data:
info.update({'ishyp': tfTOyn(data['hyperelliptic'])})
if 'hyp_involution' in data:
info.update({'hypinv': data['hyp_involution']})
gg = "/GaloisGroup/" + str(gn) + "T" + str(gt)
if 'full_auto' in data:
info.update({'fullauto': groupid_to_meaningful(data['full_auto']),
'signH':sign_display(data['signH']),
'higgenlabel' : data['full_label'] })
higgenstrg = "/HigherGenus/C/aut/" + data['full_label']
friends = [('Family of full automorphisms', higgenstrg )]
else:
friends = [ ]
bread = get_bread([(label, ' ')])
learnmore =[('Completeness of the data', url_for(".completeness_page")),
('Source of the data', url_for(".how_computed_page")),
('Labeling convention', url_for(".labels_page"))]
downloads = [('Download this example', '.')]
return render_template("hgcwa-show-curve.html", credit=HGCwA_credit,
title=title, bread=bread, info=info,
properties2=prop2, friends=friends,
learnmore=learnmore, downloads=downloads)
def render_group_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
label = label.replace('t', 'T')
data = db.gps_transitive.lookup(label)
if data is None:
bread = get_bread([("Search Error", ' ')])
info['err'] = "Group " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
data['label_raw'] = label.lower()
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(n, data['subs'])
data['resolve'] = resolve_display(data['resolve'])
if data['gapid'] == 0:
data['gapid'] = "Data 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 = wgg.arith_equivalent()
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.'
else:
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.'
if len(data['otherreps']) == 0:
data['otherreps']="There is no other low degree representation."
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({'degree': n, 'galt': 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),
('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)
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 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 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("emf.render_elliptic_modular_forms",
level=info['level'],
weight=info['weight'],
character=1))]
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'])
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 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 a Hecke Algebras in the database."
% (lab)), "error")
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']
] # getDBConnection grep make ints in database
v['disc_fac'] = [[int(i) for i in j] for j in v['disc_fac']
] # make ints in database
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("emf.render_elliptic_modular_forms",
level=info['level'],
weight=info['weight'],
character=1))]
else:
info['friends'] = []
t = "Hecke Algebra %s" % info['label']
return render_template("hecke_algebras-single.html",
info=info,
credit=credit,
title=t,
bread=bread,
properties2=info['properties'],
learnmore=learnmore_list(),
friends=info['friends'])
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", str(the_rep.conductor())),
("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()))
# 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 = {}
#mychar = the_rep.central_char()
#info['pol2']= str([((j+1),mychar(j+1, 2*the_rep.character_field())) for j in range(50)])
#info['pol3']=str(the_rep.central_character())
#info['pol3']=str(the_rep.central_char(3))
#info['pol5']=str(the_rep.central_char(5))
#info['pol7']=str(the_rep.central_char(7))
#info['pol11']=str(the_rep.central_char(11))
learnmore = [('Artin representations labels', url_for(".labels_page"))]
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)
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 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" % 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()
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.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():
# 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 = [
("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,
friends=info.pop("friends"),
learnmore=info.pop("learnmore"),
info=info,
)
def by_label(label):
clean_label = clean_input(label)
if clean_label != label:
return redirect(url_for('.by_label', label=clean_label), 301)
return search_by_label(label)
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:
query = {"label_orig": info["natural"]}
try:
parse_nf_string(info, query, "natural", name="Label", qfield="label")
return redirect(url_for(".by_label", label=clean_input(query["label"])))
except ValueError:
query["err"] = info["err"]
return search_input_error(query, bread)
query = {}
try:
parse_galgrp(info, query, qfield="galois")
parse_ints(info, query, "degree")
parse_bracketed_posints(info, query, "signature", split=False, exactlength=2)
parse_signed_ints(info, query, "discriminant", qfield=("disc_sign", "disc_abs_key"), parse_one=make_disc_key)
parse_ints(info, query, "class_number")
parse_bracketed_posints(info, query, "class_group", split=False, check_divisibility="increasing")
parse_primes(
info, query, "ur_primes", name="Unramified primes", qfield="ramps", mode="complement", to_string=True
)
if "ram_quantifier" in info and str(info["ram_quantifier"]) == "some":
mode = "append"
else:
mode = "exact"
parse_primes(info, query, "ram_primes", "ramified primes", "ramps", mode, to_string=True)
except ValueError:
return search_input_error(info, bread)
count = parse_count(info)
start = parse_start(info)
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 redirect(url_for(".by_label", clean_input(label)))
fields = C.numberfields.fields
res = fields.find(query)
if "download" in info and info["download"] != "0":
return download_search(info, res)
res = res.sort([("degree", ASC), ("disc_abs_key", ASC), ("disc_sign", ASC)])
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 number_field_search(info):
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', ' ')]
if 'natural' in info:
query = {'label_orig': info['natural']}
try:
parse_nf_string(info,query,'natural',name="Label",qfield='label')
return redirect(url_for(".by_label", label=query['label']))
except ValueError:
query['err'] = info['err']
return search_input_error(query, bread)
if 'algebra' in info:
fields=info['algebra'].split('_')
fields2=[WebNumberField.from_coeffs(a) for a in fields]
for j in range(len(fields)):
if fields2[j] is None:
fields2[j] = WebNumberField.fakenf(fields[j])
t = 'Number field algebra'
info = {}
info = {'fields': fields2}
return render_template("number_field_algebra.html", info=info, title=t, bread=bread)
query = {}
try:
parse_galgrp(info,query, qfield='galois')
parse_ints(info,query,'degree')
parse_bracketed_posints(info,query,'signature',split=False,exactlength=2)
parse_signed_ints(info,query,'discriminant',qfield=('disc_sign','disc_abs_key'),parse_one=make_disc_key)
parse_ints(info,query,'class_number')
parse_bracketed_posints(info,query,'class_group',split=False,check_divisibility='increasing')
parse_primes(info,query,'ur_primes',name='Unramified primes',qfield='ramps',mode='complement',to_string=True)
# modes are now contained (in), exactly, include
if 'ram_quantifier' in info and str(info['ram_quantifier']) == 'include':
mode = 'append'
parse_primes(info,query,'ram_primes','ramified primes','ramps',mode,to_string=True)
elif 'ram_quantifier' in info and str(info['ram_quantifier']) == 'contained':
parse_primes(info,query,'ram_primes','ramified primes','ramps_all','subsets',to_string=False)
pass # build list
else:
mode = 'liststring'
parse_primes(info,query,'ram_primes','ramified primes','ramps_all',mode)
except ValueError:
return search_input_error(info, bread)
count = parse_count(info)
start = parse_start(info)
# nf_logger.debug(query)
info['query'] = dict(query)
if 'lucky' in info:
one = nfdb().find_one(query)
if one:
label = one['label']
return redirect(url_for(".by_label", label=clean_input(label)))
fields = nfdb()
res = fields.find(query)
res = res.sort([('degree', ASC), ('disc_abs_key', ASC),('disc_sign', ASC)])
if 'download' in info and info['download'] != '0':
return download_search(info, res)
# equivalent to
# nres = res.count()
#if(start >= nres):
# start -= (1 + (start - nres) / count) * count
#if(start < 0):
# start = 0
# res = res.skip(start).limit(count)
try:
start, nres, res = search_cursor_timeout_decorator(res, start, count);
except ValueError as err:
info['err'] = err;
return search_input_error(info, bread)
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 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] = split_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_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 = hecke_l_adic_db().find_one({'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)
res = hecke_l_adic_db().find({'level': data['level'],'weight': data['weight'],'orbit_label': data['orbit_label'], 'ell': data['ell']})
info['num_l_adic_orbits']=res.count()
res_clean = []
for f in res:
f_clean = {}
f_clean['index']=int(f['index'])
f_clean['orbit_label']=str(f['orbit_label'])
f_clean['ell']=int(f['ell'])
if f['idempotent'] != "":
f['dim']=len(sage_eval(f['idempotent']))
l_max= sage_eval(f['idempotent'])[0][0].ndigits()
if f['dim']>4 or l_max>5:
f_clean['idempotent_display']=[]
elif f['dim']==1:
f_clean['idempotent_display']=sage_eval(f['idempotent'])[0][0]
else:
f_clean['idempotent_display']=latex(matrix(sage_eval(f['idempotent'])))
else:
f_clean['idempotent_display']=latex(matrix([[1]]))
f_clean['download_id'] = [(i, url_for(".render_hecke_algebras_webpage_ell_download", orbit_label=f_clean['orbit_label'], index=f_clean['index'], prime=f_clean['ell'], lang=i, obj='idempotents')) for i in ['magma','sage']] # for 'gp' the code does not work, since p-adics are not implemented
try:
f_clean['deg']=int(f['field'][1])
f_clean['field_poly']=str(f['field'][2])
f_clean['dim']=int(f['structure'][0])
f_clean['num_gen']=int(f['structure'][1])
f_clean['gens']=[[int(sage_eval(f['structure'][2]).index(i)+1), str(i)] for i in sage_eval(f['structure'][2])]
f_clean['rel']=sage_eval(f['structure'][3])
f_clean['grading']=[int(i) for i in f['properties'][0]]
f_clean['gorenstein_def']=int(f['properties'][1])
if f_clean['gorenstein_def']==0:
f_clean['gorenstein']="yes"
else:
f_clean['gorenstein']="no"
f_clean['operators_mod_l']=[[int(i) for i in j] for j in f['operators']]
f_clean['num_hecke_op']=len(f_clean['operators_mod_l'])
f_clean['download_op'] = [(i, url_for(".render_hecke_algebras_webpage_ell_download", orbit_label=f_clean['orbit_label'], index=f_clean['index'], prime=f_clean['ell'], lang=i, obj='operators')) for i in ['magma','sage']] # for 'gp' the code does not work
f_clean['size_op']=sqrt(len(f_clean['operators_mod_l'][0]))
if f_clean['size_op']>4:
f_clean['operators_mod_l_display']=[]
elif f_clean['size_op']==1:
f_clean['operators_mod_l_display']=[[i+1, f_clean['operators_mod_l'][i][0]] for i in range(0,10)]
else:
f_clean['operators_mod_l_display']=[[i+1,latex(matrix(f_clean['size_op'],f_clean['size_op'], f_clean['operators_mod_l'][i]))] for i in range(0,5)]
res_clean.append(f_clean)
except KeyError:
pass
info['l_adic_orbits']=res_clean
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("emf.render_elliptic_modular_forms", level=info['level'], weight=info['weight'], character=1))]
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'])
def hgm_search(info):
#info = to_dict(args)
bread = get_bread([("Search Results", '')])
query = {}
queryab = {}
queryabrev = {}
if 'jump_to' in 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"))
family_search = False
if info.get('Submit Family') or info.get('family'):
family_search = True
# generic, irreducible not in DB yet
try:
parse_ints(info, query, 'degree')
parse_ints(info, query, 'weight')
parse_bracketed_posints(info,
query,
'famhodge',
'family Hodge vector',
split=False)
parse_restricted(info,
query,
'sign',
allowed=['+1', 1, -1],
process=int)
for param in [
'A', 'B', 'A2', 'B2', 'A3', 'B3', 'A5', 'B5', 'A7', 'B7',
'Au2', 'Bu2', 'Au3', 'Bu3', 'Au5', 'Bu5', 'Au7', 'Bu7'
]:
parse_bracketed_posints(
info,
queryab,
param,
split=False,
listprocess=lambda a: sorted(a, reverse=True))
# Make a version to search reversed way
if not family_search:
parse_ints(info, query, 'conductor', 'Conductor', 'cond')
parse_rational(info, query, 't')
parse_bracketed_posints(info, query, 'hodge', 'Hodge vector')
except ValueError:
if family_search:
return render_template(
"hgm-search.html",
info=info,
title="Hypergeometric Family over $\Q$ Search Result",
bread=bread,
credit=HGM_credit,
learnmore=learnmore_list())
return render_template(
"hgm-search.html",
info=info,
title="Hypergeometric Motive over $\Q$ Search Result",
bread=bread,
credit=HGM_credit,
learnmore=learnmore_list())
# Now combine the parts of the query if there are A,B parts
if queryab != {}:
for k in queryab.keys():
queryabrev[k + 'rev'] = queryab[k]
queryab.update(query)
queryabrev.update(query)
query = {'$or': [queryab, queryabrev]}
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:
#query['leader'] = '1'
res = familydb().find(query).sort([('label', pymongo.ASCENDING)])
else:
res = motivedb().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,
learnmore=learnmore_list())
else:
return render_template(
"hgm-search.html",
info=info,
title="Hypergeometric Motive over $\Q$ Search Result",
bread=bread,
credit=HGM_credit,
learnmore=learnmore_list())
def render_artin_representation_webpage(label):
if re.compile(r'^\d+$').match(label):
return artin_representation_search(**{'dimension': label})
bread = get_bread([(label, ' ')])
# label=dim.cond.nTt.indexcj, c is literal, j is index in conj class
# Should we have a big try around this to catch bad labels?
clean_label = clean_input(label)
if clean_label != label:
return redirect(url_for('.render_artin_representation_webpage', label=clean_label), 301)
try:
the_rep = ArtinRepresentation(label)
except:
flash(Markup("Error: <span style='color:black'>%s</span> is not the label of an Artin representation in the database." % (label)), "error")
return search_input_error({'err':''}, bread)
extra_data = {} # for testing?
extra_data['galois_knowl'] = group_display_knowl(5,3) # for testing?
#artin_logger.info("Found %s" % (the_rep._data))
title = "Artin Representation %s" % label
the_nf = the_rep.number_field_galois_group()
if the_rep.sign() == 0:
processed_root_number = "not computed"
else:
processed_root_number = str(the_rep.sign())
properties = [("Label", label),
("Dimension", str(the_rep.dimension())),
("Group", the_rep.group()),
#("Conductor", str(the_rep.conductor())),
("Conductor", "$" + 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()))
# 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={}
#mychar = the_rep.central_char()
#info['pol2']= str([((j+1),mychar(j+1, 2*the_rep.character_field())) for j in range(50)])
#info['pol3']=str(the_rep.central_character())
#info['pol3']=str(the_rep.central_char(3))
#info['pol5']=str(the_rep.central_char(5))
#info['pol7']=str(the_rep.central_char(7))
#info['pol11']=str(the_rep.central_char(11))
learnmore=[('Artin representations labels', url_for(".labels_page"))]
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)
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 hgm_search(info):
#info = to_dict(args)
bread = get_bread([("Search Results", '')])
query = {}
queryab = {}
queryabrev = {}
if 'jump_to' in 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"))
family_search = False
if info.get('Submit Family') or info.get('family'):
family_search = True
# generic, irreducible not in DB yet
try:
parse_ints(info, query, 'degree')
parse_ints(info, query, 'weight')
parse_bracketed_posints(info, query, 'famhodge', 'family Hodge vector',split=False)
parse_restricted(info, query, 'sign', allowed=['+1',1,-1], process=int)
for param in ['A', 'B', 'A2', 'B2', 'A3', 'B3', 'A5', 'B5', 'A7', 'B7',
'Au2', 'Bu2', 'Au3', 'Bu3', 'Au5', 'Bu5', 'Au7', 'Bu7']:
parse_bracketed_posints(info, queryab, param, split=False,
listprocess=lambda a: sorted(a, reverse=True))
# Make a version to search reversed way
if not family_search:
parse_ints(info, query, 'conductor', 'Conductor' , 'cond')
parse_rational(info, query, 't')
parse_bracketed_posints(info, query, 'hodge', 'Hodge vector')
except ValueError:
if family_search:
return render_template("hgm-search.html", info=info, title="Hypergeometric Family over $\Q$ Search Result", bread=bread, credit=HGM_credit, learnmore=learnmore_list())
return render_template("hgm-search.html", info=info, title="Hypergeometric Motive over $\Q$ Search Result", bread=bread, credit=HGM_credit, learnmore=learnmore_list())
# Now combine the parts of the query if there are A,B parts
if queryab != {}:
for k in queryab.keys():
queryabrev[k+'rev'] = queryab[k]
queryab.update(query)
queryabrev.update(query)
query = {'$or':[queryab, queryabrev]}
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:
#query['leader'] = '1'
res = familydb().find(query).sort([('label', pymongo.ASCENDING)])
else:
res = motivedb().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, learnmore=learnmore_list())
else:
return render_template("hgm-search.html", info=info, title="Hypergeometric Motive over $\Q$ Search Result", bread=bread, credit=HGM_credit, learnmore=learnmore_list())
def render_artin_representation_webpage(label):
if re.compile(r'^\d+$').match(label):
return artin_representation_search(**{'dimension': label})
bread = get_bread([(label, ' ')])
# label=dim.cond.nTt.indexcj, c is literal, j is index in conj class
# Should we have a big try around this to catch bad labels?
label = clean_input(label)
try:
the_rep = ArtinRepresentation(label)
except:
flash(Markup("Error: <span style='color:black'>%s</span> is not the label of an Artin representation in the database." % (label)), "error")
return search_input_error({'err':''}, bread)
extra_data = {} # for testing?
C = getDBConnection()
extra_data['galois_knowl'] = group_display_knowl(5,3,C) # for testing?
#artin_logger.info("Found %s" % (the_rep._data))
title = "Artin representation %s" % label
the_nf = the_rep.number_field_galois_group()
from lmfdb.number_field_galois_groups import nfgg_page
from lmfdb.number_field_galois_groups.main import by_data
if the_rep.sign() == 0:
processed_root_number = "not computed"
else:
processed_root_number = str(the_rep.sign())
properties = [("Dimension", str(the_rep.dimension())),
("Group", the_rep.group()),
#("Conductor", str(the_rep.conductor())),
("Conductor", "$" + 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 cc.modulus <= 100000:
if the_rep.dimension()==1:
friends.append(("Corresponding Dirichlet character", url_for("characters.render_Dirichletwebpage", modulus=cc.modulus, number=cc.number)))
else:
friends.append(("Determinant character", url_for("characters.render_Dirichletwebpage", modulus=cc.modulus, number=cc.number)))
# once the L-functions are in the database, the link can always be shown
if the_rep.dimension() <= 6:
friends.append(("L-function", url_for("l_functions.l_function_artin_page",
label=the_rep.label())))
info={}
#mychar = the_rep.central_char()
#info['pol2']= str([((j+1),mychar(j+1, 2*the_rep.character_field())) for j in range(50)])
#info['pol3']=str(the_rep.central_character())
#info['pol3']=str(the_rep.central_char(3))
#info['pol5']=str(the_rep.central_char(5))
#info['pol7']=str(the_rep.central_char(7))
#info['pol11']=str(the_rep.central_char(11))
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)
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", ' ')])
info['err'] = "Group " + label + " was not found in the database."
info['label'] = label
return search_input_error(info, bread)
data['label_raw'] = label.lower()
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 data['gapid'] == 0:
data['gapid'] = "Data not available"
else:
data['gapid'] = small_group_display_knowl(int(data['order']),int(data['gapid']),C, str([int(data['order']),int(data['gapid'])]))
data['otherreps'] = wgg.otherrep_list()
ae = wgg.arith_equivalent()
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.'
else:
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.'
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 = [('Label', label),
('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 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 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 a Hecke Algebras in the database." % (lab)),"error")
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, properties2=info['properties'], learnmore=learnmore_list(), friends=info['friends'])
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, ' ')])
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_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' % 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()
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.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():
# 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 = [('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, friends=info.pop('friends'), learnmore=info.pop('learnmore'), info=info)
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 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:
query = {'label_orig': info['natural']}
try:
parse_nf_string(info,query,'natural',name="Label",qfield='label')
return redirect(url_for(".by_label", label= clean_input(query['label'])))
except ValueError:
query['err'] = info['err']
return search_input_error(query, bread)
query = {}
try:
parse_galgrp(info,query, qfield='galois')
parse_ints(info,query,'degree')
parse_bracketed_posints(info,query,'signature',split=False,exactlength=2)
parse_signed_ints(info,query,'discriminant',qfield=('disc_sign','disc_abs_key'),parse_one=make_disc_key)
parse_ints(info,query,'class_number')
parse_bracketed_posints(info,query,'class_group',split=False,check_divisibility='increasing')
parse_primes(info,query,'ur_primes',name='Unramified primes',qfield='ramps',mode='complement',to_string=True)
if 'ram_quantifier' in info and str(info['ram_quantifier']) == 'some':
mode = 'append'
else:
mode = 'exact'
parse_primes(info,query,'ram_primes','ramified primes','ramps',mode,to_string=True)
except ValueError:
return search_input_error(info, bread)
count = parse_count(info)
start = parse_start(info)
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 redirect(url_for(".by_label", clean_input(label)))
fields = C.numberfields.fields
res = fields.find(query)
if 'download' in info and info['download'] != '0':
return download_search(info, res)
res = res.sort([('degree', ASC), ('disc_abs_key', ASC),('disc_sign', ASC)])
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 = hecke_l_adic_db().find_one({'orbit_label': lab , 'ell': ell})
if data is None:
t = "Hecke Agebras 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)
res = hecke_l_adic_db().find({'level': data['level'],'weight': data['weight'],'orbit_label': data['orbit_label'], 'ell': data['ell']})
info['num_l_adic_orbits']=res.count()
res_clean = []
for f in res:
f_clean = {}
f_clean['index']=int(f['index'])
f_clean['orbit_label']=str(f['orbit_label'])
f_clean['ell']=int(f['ell'])
if f['idempotent'] != "":
f['dim']=len(sage_eval(f['idempotent']))
l_max= sage_eval(f['idempotent'])[0][0].ndigits()
if f['dim']>4 or l_max>5:
f_clean['idempotent_display']=[]
elif f['dim']==1:
f_clean['idempotent_display']=sage_eval(f['idempotent'])[0][0]
else:
f_clean['idempotent_display']=latex(matrix(sage_eval(f['idempotent'])))
else:
f_clean['idempotent_display']=latex(matrix([[1]]))
f_clean['download_id'] = [(i, url_for(".render_hecke_algebras_webpage_ell_download", orbit_label=f_clean['orbit_label'], index=f_clean['index'], prime=f_clean['ell'], lang=i, obj='idempotents')) for i in ['magma','sage']] # for 'gp' the code does not work, since p-adics are not implemented
try:
f_clean['deg']=int(f['field'][1])
f_clean['field_poly']=str(f['field'][2])
f_clean['dim']=int(f['structure'][0])
f_clean['num_gen']=int(f['structure'][1])
f_clean['gens']=[[int(sage_eval(f['structure'][2]).index(i)+1), str(i)] for i in sage_eval(f['structure'][2])]
f_clean['rel']=sage_eval(f['structure'][3])
f_clean['grading']=[int(i) for i in f['properties'][0]]
f_clean['gorenstein_def']=int(f['properties'][1])
if f_clean['gorenstein_def']==0:
f_clean['gorenstein']="yes"
else:
f_clean['gorenstein']="no"
f_clean['operators_mod_l']=[[int(i) for i in j] for j in f['operators']]
f_clean['num_hecke_op']=len(f_clean['operators_mod_l'])
f_clean['download_op'] = [(i, url_for(".render_hecke_algebras_webpage_ell_download", orbit_label=f_clean['orbit_label'], index=f_clean['index'], prime=f_clean['ell'], lang=i, obj='operators')) for i in ['magma','sage']] # for 'gp' the code does not work
f_clean['size_op']=sqrt(len(f_clean['operators_mod_l'][0]))
if f_clean['size_op']>4:
f_clean['operators_mod_l_display']=[]
elif f_clean['size_op']==1:
f_clean['operators_mod_l_display']=[[i+1, f_clean['operators_mod_l'][i][0]] for i in range(0,10)]
else:
f_clean['operators_mod_l_display']=[[i+1,latex(matrix(f_clean['size_op'],f_clean['size_op'], f_clean['operators_mod_l'][i]))] for i in range(0,5)]
res_clean.append(f_clean)
except KeyError:
pass
info['l_adic_orbits']=res_clean
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("emf.render_elliptic_modular_forms", level=info['level'], weight=info['weight'], character=1))]
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'])
def render_family(args):
info = {}
if 'label' in args:
label = clean_input(args['label'])
dataz = list(db.hgcwa_passports.search({'label':label}))
if len(dataz) == 0:
flash_error( "No family with label %s was found in the database.", label)
return redirect(url_for(".index"))
data=dataz[0]
g = data['genus']
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', '\(%d\)' % g),
('Group', '\(%s\)' % pretty_group),
('Signature', '\(%s\)' % 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 = [("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")),
('Labeling convention', url_for(".labels_page"))]
downloads = [('Download Magma code', url_for(".hgcwa_code_download", label=label, download_type='magma')),
('Download Gap code', url_for(".hgcwa_code_download", label=label, download_type='gap'))]
return render_template("hgcwa-show-family.html",
title=title, bread=bread, info=info,
properties2=prop2, friends=friends,
learnmore=learnmore, downloads=downloads, credit=credit)
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 len(dataz) == 0:
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']
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', '\(%d\)' % g),
('Small Group', '\(%s\)' % pretty_group),
('Signature', '\(%s\)' % sign_display(ast.literal_eval(data['signature']))),
('Generating Vectors','\(%d\)' % 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)
})
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=[]
for perm in dat['gen_vectors']:
cycperm=Permutation(perm).cycle_string()
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, refined_p = split_passport_label(label)
if Lfriends:
for Lf in Lfriends:
friends = [("Full automorphism " + Lf, Lf),("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")),
('Labeling convention', url_for(".labels_page"))]
downloads = [('Download Magma code', url_for(".hgcwa_code_download", label=label, download_type='magma')),
('Download Gap code', url_for(".hgcwa_code_download", label=label, download_type='gap'))]
return render_template("hgcwa-show-passport.html",
title=title, bread=bread, info=info,
properties2=prop2, friends=friends,
learnmore=learnmore, downloads=downloads, credit=credit)
def render_lattice_webpage(**args):
data = 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)
data = lattice_db().find_one({'$or':[{'label': lab }, {'name': lab }]})
if data is None:
t = "Integral Lattices Search Error"
bread = [('Lattice', 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(data)
info['friends'] = []
bread = [('Lattice', url_for(".lattice_render_webpage")), ('%s' % data['label'], ' ')]
credit = lattice_credit
f = lattice_db().find_one({'dim': data['dim'],'det': data['det'],'level': data['level'],'gram': data['gram'],'minimum': data['minimum'],'class_number': data['class_number'],'aut': data[ 'aut'],'name': data['name']})
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())
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 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:
query = {'label_orig': info['natural']}
try:
parse_nf_string(info,query,'natural',name="Label",qfield='label')
return redirect(url_for(".by_label", label= clean_input(query['label_orig'])))
except ValueError:
query['err'] = info['err']
return search_input_error(query, bread)
query = {}
try:
parse_galgrp(info,query, qfield='galois')
parse_ints(info,query,'degree')
parse_bracketed_posints(info,query,'signature',split=False,exactlength=2)
parse_signed_ints(info,query,'discriminant',qfield=('disc_sign','disc_abs_key'),parse_one=make_disc_key)
parse_ints(info,query,'class_number')
parse_bracketed_posints(info,query,'class_group',split=False,check_divisibility='increasing')
parse_primes(info,query,'ur_primes',name='Unramified primes',qfield='ramps',mode='complement',to_string=True)
# modes are now contained (in), exactly, include
if 'ram_quantifier' in info and str(info['ram_quantifier']) == 'include':
mode = 'append'
parse_primes(info,query,'ram_primes','ramified primes','ramps',mode,to_string=True)
elif 'ram_quantifier' in info and str(info['ram_quantifier']) == 'contained':
parse_primes(info,query,'ram_primes','ramified primes','ramps_all','subsets',to_string=False)
pass # build list
else:
mode = 'liststring'
parse_primes(info,query,'ram_primes','ramified primes','ramps_all',mode)
except ValueError:
return search_input_error(info, bread)
count = parse_count(info)
start = parse_start(info)
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 redirect(url_for(".by_label", label=clean_input(label)))
fields = C.numberfields.fields
res = fields.find(query)
res = res.sort([('degree', ASC), ('disc_abs_key', ASC),('disc_sign', 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_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)
data['label_raw'] = label.lower()
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_passport(args):
info = {}
if 'passport_label' in args:
label = clean_input(args['passport_label'])
C = base.getDBConnection()
dataz = C.curve_automorphisms.passports.find({'passport_label': label})
if dataz.count() is 0:
bread = get_bread([("Search error", url_for('.search'))])
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']
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 = dataz.count()
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', '\(%d\)' % g), ('Small Group', '\(%s\)' % pretty_group),
('Signature',
'\(%s\)' % sign_display(ast.literal_eval(data['signature']))),
('Generating Vectors', '\(%d\)' % 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)
})
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 = []
for perm in dat['gen_vectors']:
cycperm = Permutation(perm).cycle_string()
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, refined_p = split_passport_label(label)
if Lfriends:
for Lf in Lfriends:
friends = [("Full automorphism " + Lf, Lf),
("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")),
('Labeling convention', url_for(".labels_page"))]
downloads = [('Download Magma code',
url_for(".hgcwa_code_download",
label=label,
download_type='magma')),
('Download Gap code',
url_for(".hgcwa_code_download",
label=label,
download_type='gap'))]
return render_template("hgcwa-show-passport.html",
title=title,
bread=bread,
info=info,
properties2=prop2,
friends=friends,
learnmore=learnmore,
downloads=downloads,
credit=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:
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$'%Qp),
'hw': data['hw'],
'slopes': show_slopes(data['slopes']),
'gal': group_display_knowl(gn, gt),
'gt': gt,
'inertia': group_display_inertia(data['inertia']),
'unram': unramp,
'eisen': eisenp,
'gms': data['gms'],
'gsm': gsm,
'galphrase': galphrase,
'autstring': autstring,
'subfields': format_subfields(data['subfields'],p),
'aut': data['aut'],
})
friends = [('Galois group', "/GaloisGroup/%dT%d" % (gn, gt))]
if unramfriend != '':
friends.append(('Unramified subfield', unramfriend))
if rffriend != '':
friends.append(('Discriminant root field', rffriend))
bread = get_bread([(label, ' ')])
learnmore = [('Completeness of the data', url_for(".completeness_page")),
('Source of the data', url_for(".how_computed_page")),
('Local field labels', url_for(".labels_page"))]
return render_template("lf-show-field.html", credit=LF_credit, title=title, bread=bread, info=info, properties2=prop2, friends=friends, learnmore=learnmore)
def render_field_webpage(args):
data = None
C = base.getDBConnection()
info = {}
bread = [('Global Number Fields', url_for(".number_field_render_webpage"))]
# This function should not be called unless label is set.
label = clean_input(args['label'])
nf = WebNumberField(label)
data = {}
if nf.is_null():
bread.append(('Search results', ' '))
info['err'] = 'There is no field with label %s in the database' % label
info['label'] = args['label_orig'] if 'label_orig' in args else args['label']
return search_input_error(info, bread)
info['wnf'] = nf
data['degree'] = nf.degree()
data['class_number'] = nf.class_number()
t = nf.galois_t()
n = nf.degree()
data['is_galois'] = nf.is_galois()
data['is_abelian'] = nf.is_abelian()
if nf.is_abelian():
conductor = nf.conductor()
data['conductor'] = conductor
dirichlet_chars = nf.dirichlet_group()
if len(dirichlet_chars)>0:
data['dirichlet_group'] = ['<a href = "%s">$\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 render_hgm_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
data = motivedb().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']
myfam = familydb().find_one({'A': A, 'B': B})
if myfam is None:
det = 'data not computed'
else:
det = myfam['det']
det = [det[0],str(det[1])]
d1 = det[1]
d1 = re.sub(r'\s','', d1)
d1 = re.sub(r'(.)\(', r'\1*(', d1)
R = PolynomialRing(ZZ, 't')
if det[1]=='':
d2 = R(1)
else:
d2 = R(d1)
det = d2(QQ(data['t']))*det[0]
t = latex(QQ(data['t']))
typee = 'Orthogonal'
if (data['weight'] % 2) == 1 and (data['degree'] % 2) == 0:
typee = 'Symplectic'
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])
locinfo[j][2] = list_to_factored_poly_otherorder(locinfo[j][2], vari='x')
hodge = string2list(data['hodge'])
famhodge = string2list(data['famhodge'])
prop2 = [
('Degree', '\(%s\)' % data['degree']),
('Weight', '\(%s\)' % data['weight']),
('Hodge vector', '\(%s\)' % hodge),
('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,
'famhodge': famhodge,
'cond': data['cond'],
'req': data['req'],
'lcms': data['lcms'],
'type': typee,
'det': det,
'locinfo': locinfo
})
AB_data, t_data = data["label"].split("_t")
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, learnmore=learnmore_list())
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 render_hgm_family_webpage(args):
data = None
info = {}
if 'label' in args:
label = clean_input(args['label'])
data = familydb().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 = string2list(data['A'])
B = string2list(data['B'])
hodge = data['famhodge']
mydet = data['det']
detexp = QQ(data['weight']*data['degree'])
detexp = -detexp/2
mydet = r'\Q(%s)\otimes\Q(\sqrt{'%str(detexp)
if int(data['det'][0]) != 1:
mydet += str(data['det'][0])
if len(data['det'][1])>0:
mydet += data['det'][1]
if int(data['det'][0]) == 1 and len(data['det'][1])==0:
mydet += '1'
mydet += '})'
bezoutmat = matrix(data['bezout'])
bezoutdet = bezoutmat.det()
bezoutmat = latex(bezoutmat)
snf = string2list(data['snf'])
snf = list2Cnstring(snf)
typee = 'Orthogonal'
if (data['weight'] % 2) == 1 and (data['degree'] % 2) == 0:
typee = 'Symplectic'
ppart = [[2, [string2list(u) for u in [data['A2'],data['B2'],data['C2']]]],
[3, [string2list(u) for u in [data['A3'],data['B3'],data['C3']]]],
[5, [string2list(u) for u in [data['A5'],data['B5'],data['C5']]]],
[7, [string2list(u) for u in [data['A7'],data['B7'],data['C7']]]]]
prop2 = [
('Degree', '\(%s\)' % data['degree']),
('Weight', '\(%s\)' % data['weight'])
]
mono = [[m[0], dogapthing(m[1]),
getgroup(m[1],m[0]),
latex(ZZ(m[1][0]).factor())] for m in data['mono']]
mono = [[m[0], m[1], m[2][0], splitint(m[1][0]/m[2][1],m[0]), m[3]] for m in mono]
info.update({
'A': A,
'B': B,
'degree': data['degree'],
'weight': data['weight'],
'hodge': hodge,
'det': mydet,
'snf': snf,
'bezoutmat': bezoutmat,
'bezoutdet': bezoutdet,
'mono': mono,
'imprim': data['imprim'],
'ppart': ppart,
'type': typee,
'junk': small_group_display_knowl(18,2,db()),
'showlist': showlist
})
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, learnmore=learnmore_list())
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 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 = hecke_db().find_one({'label': lab })
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 a Hecke Algebras in the database." % (lab)),"error")
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
f = hecke_db().find_one({'level': data['level'],'weight': data['weight'],'num_orbits': data['num_orbits']})
info['level']=int(f['level'])
info['weight']= int(f['weight'])
info['num_orbits']= int(f['num_orbits'])
dim_count = "not available"
orb = hecke_orbits_db().find({'parent_label': f['label']}).sort([('orbit', ASC)])
if orb.count()!=0:
#consistency check
if orb.count()!= int(f['num_orbits']):
return search_input_error(info)
dim_count=0
res_clean = []
for v in orb:
v_clean = {}
v_clean['orbit_label'] = v['orbit_label']
v_clean['hecke_op'] = v['hecke_op']
v_clean['gen'] = v['gen']
v_clean['dim'] = int(matrix(sage_eval(v_clean['hecke_op'])[0]).dimensions()[0])
dim_count = dim_count + v_clean['dim']
if v_clean['dim']>4:
v_clean['hecke_op_display'] = []
elif v_clean['dim']==1:
v_clean['hecke_op_display'] = [[i+1, (sage_eval(v_clean['hecke_op'])[i])[0][0]] for i in range(0,10)]
else:
v_clean['hecke_op_display'] = [[i+1, latex(matrix(sage_eval(v_clean['hecke_op'])[i]))] for i in range(0,5)]
v_clean['num_hecke_op'] = v['num_hecke_op']
v_clean['download_op'] = [(i, url_for(".render_hecke_algebras_webpage_download", orbit_label=v_clean['orbit_label'], lang=i, obj='operators')) for i in ['gp', 'magma','sage']]
if 'Zbasis' in v.keys():
v_clean['Zbasis'] = [[int(i) for i in j] for j in v['Zbasis']]
if v_clean['dim']>4:
v_clean['gen_display'] = []
elif v_clean['dim']==1:
v_clean['gen_display'] = [v_clean['Zbasis'][0][0]]
else:
v_clean['gen_display'] = [latex(matrix(v_clean['dim'],v_clean['dim'], v_clean['Zbasis'][i])) for i in range(0,v_clean['dim'])]
v_clean['discriminant'] = int(v['discriminant'])
v_clean['disc_fac'] = [[int(i) for i in j] for j in v['disc_fac']]
v_clean['Qbasis'] = [int(i) for i in v['Qbasis']]
v_clean['Qalg_gen'] = [int(i) for i in v['Qalg_gen']]
if 'inner_twists' in v.keys():
v_clean['inner_twists'] = [str(i) for i in v['inner_twists']]
else:
v_clean['inner_twists'] = "not available"
v_clean['download_gen'] = [(i, url_for(".render_hecke_algebras_webpage_download", orbit_label=v_clean['orbit_label'], lang=i, obj='gen')) for i in ['gp', 'magma','sage']]
res_clean.append(v_clean)
info['orbits'] = res_clean
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("emf.render_elliptic_modular_forms", level=info['level'], weight=info['weight'], character=1))]
else:
info['friends'] = []
t = "Hecke Algebra %s" % info['label']
return render_template("hecke_algebras-single.html", info=info, credit=credit, title=t, bread=bread, properties2=info['properties'], learnmore=learnmore_list(), friends=info['friends'])