def render_st_group(info, portrait=None):
""" render html page for Sato-Tate group described by info """
prop2 = [('Label', '%s' % info['label'])]
if portrait:
prop2 += [
(None,
' <img src="%s" width="220" height="124"/>' % portrait)
]
prop2 += [
('Name', r'\(%s\)' % info['pretty']),
('Weight', prop_int_pretty(info['weight'])),
('Degree', prop_int_pretty(info['degree'])),
('Real dimension', prop_int_pretty(info['real_dimension'])),
('Components', prop_int_pretty(info['components'])),
('Contained in', r'\(%s\)' % info['ambient']),
('Identity Component', r'\(%s\)' % info['identity_component']),
('Component group', r'\(%s\)' % info['component_group']),
]
bread = get_bread([
('Weight %d' % info['weight'],
url_for('.index') + '?weight=' + str(info['weight'])),
('Degree %d' % info['degree'], url_for('.index') + '?weight=' +
str(info['weight']) + '°ree=' + str(info['degree'])),
(info['name'], '')
])
title = r'Sato-Tate group \(' + info['pretty'] + r'\) of Weight %d' % info[
'weight'] + ' and Degree %d' % info['degree']
return render_template('st_display.html',
properties=prop2,
credit=credit_string,
info=info,
bread=bread,
learnmore=learnmore_list(),
title=title,
KNOWL_ID='st_group.%s' % (info['label']))
def properties(self):
f = [("Label", [self.label])]
f.extend([("Modulus", [prop_int_pretty(self.modulus)]),
("Conductor", [prop_int_pretty(self.conductor)]),
("Order", [prop_int_pretty(self.order)]),
("Real", [self.isreal]), ("Primitive", [self.isprimitive])])
if self.isminimal:
f.append(("Minimal", [self.isminimal]))
if self.parity:
f.append(("Parity", [self.parity]))
return f
def properties(self):
props = [
(None,
'<img src="{0}" width="200" height="150" style="margin:10px;"/>'.
format(self.portrait))
] if self.portrait is not None else []
props += [
('Level', prop_int_pretty(self.level)),
('Weight', prop_int_pretty(self.weight)),
('Character', self.character_label),
('Symmetry', self.symmetry_pretty),
]
if self.conrey_index == 1:
props.append(('Fricke sign', self.fricke_pretty))
return props
def render_hmf_webpage(**args):
if 'data' in args:
data = args['data']
label = data['label']
else:
label = str(args['label'])
data = get_hmf(label)
if data is None:
flash_error(
"%s is not a valid Hilbert modular form label. It must be of the form (number field label) - (level label) - (orbit label) separated by dashes, such as 2.2.5.1-31.1-a",
args['label'])
return search_input_error()
info = {}
try:
info['count'] = args['count']
except KeyError:
info['count'] = 50
hmf_field = get_hmf_field(data['field_label'])
gen_name = findvar(hmf_field['ideals'])
nf = WebNumberField(data['field_label'], gen_name=gen_name)
info['hmf_field'] = hmf_field
info['field'] = nf
info['base_galois_group'] = nf.galois_string()
info['field_degree'] = nf.degree()
info['field_disc'] = str(nf.disc())
info['field_poly'] = teXify_pol(str(nf.poly()))
info.update(data)
info['downloads'] = [('Modular form to Magma',
url_for(".render_hmf_webpage_download",
field_label=info['field_label'],
label=info['label'],
download_type='magma')),
('Eigenvalues to Sage',
url_for(".render_hmf_webpage_download",
field_label=info['field_label'],
label=info['label'],
download_type='sage'))]
# figure out friends
# first try to see if there is an instance of this HMF on Lfun db
url = 'ModularForm/GL2/TotallyReal/{}/holomorphic/{}'.format(
info['field_label'], info['label'])
Lfun = get_lfunction_by_url(url)
if Lfun:
instances = get_instances_by_Lhash_and_trace_hash(
Lfun['Lhash'], Lfun['degree'], Lfun['trace_hash'])
# This will also add the EC/G2C, as this how the Lfun was computed
info['friends'] = names_and_urls(instances, exclude={url})
info['friends'] += [('L-function',
url_for("l_functions.l_function_hmf_page",
field=info['field_label'],
label=info['label'],
character='0',
number='0'))]
else:
# if there is no instance
# old code
if hmf_field['narrow_class_no'] == 1 and nf.disc(
)**2 * data['level_norm'] < 40000:
info['friends'] = [('L-function',
url_for("l_functions.l_function_hmf_page",
field=info['field_label'],
label=info['label'],
character='0',
number='0'))]
else:
info['friends'] = [('L-function not available', "")]
if data['dimension'] == 1: # Try to attach associated elliptic curve
lab = split_class_label(info['label'])
ec_from_hmf = db.ec_nfcurves.lookup(label + '1')
if ec_from_hmf is None:
info['friends'] += [('Elliptic curve not available', "")]
else:
info['friends'] += [('Isogeny class ' + info['label'],
url_for("ecnf.show_ecnf_isoclass",
nf=lab[0],
conductor_label=lab[1],
class_label=lab[2]))]
bread = get_bread(data["label"])
t = "Hilbert cusp form %s" % info['label']
forms_dims = db.hmf_forms.search(
{
'field_label': data['field_label'],
'level_ideal': data['level_ideal']
},
projection='dimension')
info['newspace_dimension'] = sum(forms_dims)
# Get hecke_polynomial, hecke_eigenvalues and AL_eigenvalues
try:
numeigs = request.args['numeigs']
numeigs = int(numeigs)
except:
numeigs = 20
info['numeigs'] = numeigs
hecke_pol = data['hecke_polynomial']
eigs = [str(eig) for eig in data['hecke_eigenvalues']]
eigs = eigs[:min(len(eigs), numeigs)]
AL_eigs = data['AL_eigenvalues']
primes = hmf_field['primes']
n = min(len(eigs), len(primes))
info['eigs'] = [{
'eigenvalue': add_space_if_positive(teXify_pol(eigs[i])),
'prime_ideal': teXify_pol(primes[i]),
'prime_norm': primes[i][1:primes[i].index(',')]
} for i in range(n)]
try:
display_eigs = request.args['display_eigs']
if display_eigs in ['True', 'true', '1', 'yes']:
display_eigs = True
else:
display_eigs = False
except KeyError:
display_eigs = False
if 'numeigs' in request.args:
display_eigs = True
info['hecke_polynomial'] = r"\(" + teXify_pol(hecke_pol) + r"\)"
if not AL_eigs: # empty list
if data['level_norm'] == 1: # OK, no bad primes
info['AL_eigs'] = 'none'
else: # not OK, AL eigs are missing
info['AL_eigs'] = 'missing'
else:
info['AL_eigs'] = [{
'eigenvalue': teXify_pol(al[1]),
'prime_ideal': teXify_pol(al[0]),
'prime_norm': al[0][1:al[0].index(',')]
} for al in data['AL_eigenvalues']]
max_eig_len = max([len(eig['eigenvalue']) for eig in info['eigs']])
display_eigs = display_eigs or (max_eig_len <= 300)
info['display_eigs'] = display_eigs
if not display_eigs:
for eig in info['eigs']:
if len(eig['eigenvalue']) > 300:
eig['eigenvalue'] = '...'
info['level_ideal'] = teXify_pol(info['level_ideal'])
if 'is_CM' in data:
is_CM = data['is_CM']
else:
is_CM = '?'
info['is_CM'] = is_CM
if 'is_base_change' in data:
is_base_change = data['is_base_change']
else:
is_base_change = '?'
info['is_base_change'] = is_base_change
if 'q_expansions' in data:
info['q_expansions'] = data['q_expansions']
properties = [('Label', '%s' % data['label']),
('Base field', '%s' % info['field'].field_pretty()),
('Weight', '$%s$' % data['weight']),
('Level norm', prop_int_pretty(data['level_norm'])),
('Level', '$' + teXify_pol(data['level_ideal']) + '$'),
('Dimension', prop_int_pretty(data['dimension'])),
('CM', is_CM), ('Base change', is_base_change)]
return render_template(
"hilbert_modular_form.html",
downloads=info["downloads"],
info=info,
properties=properties,
credit=hmf_credit,
title=t,
bread=bread,
friends=info['friends'],
learnmore=learnmore_list(),
KNOWL_ID="mf.hilbert.%s" % label,
)
def render_sample_page(family, sam, args, bread):
info = {
'args': to_dict(args),
'sam': sam,
'latex': latex,
'type': sam.type(),
'name': sam.name(),
'full_name': sam.full_name(),
'weight': sam.weight(),
'fdeg': sam.degree_of_field(),
'is_eigenform': sam.is_eigenform(),
'field_poly': sam.field_poly()
}
if sam.is_integral() is not None:
info['is_integral'] = sam.is_integral()
if 'Sp4Z' in sam.collection():
info['space_url'] = url_for('.Sp4Z_j_space', k=info['weight'], j=0)
if 'Sp4Z_2' in sam.collection():
info['space_url'] = url_for('.Sp4Z_j_space', k=info['weight'], j=2)
info['space'] = '$' + family.latex_name.replace(
'k', '{' + str(sam.weight()) + '}') + '$'
if 'space_url' in info:
bread.append((info['space'], info['space_url']))
info['space_href'] = '<a href="%s">%s</d>' % (
info['space_url'],
info['space']) if 'space_url' in info else info['space']
if info['field_poly'].disc() < 10**10:
label = poly_to_field_label(info['field_poly'])
if label:
info['field_label'] = label
info['field_url'] = url_for('number_fields.by_label', label=label)
info['field_href'] = '<a href="%s">%s</a>' % (info['field_url'],
field_pretty(label))
bread.append((info['name'], ''))
title = 'Siegel modular forms sample ' + info['full_name']
properties = [('Space', info['space_href']), ('Name', info['name']),
('Type', '<br>'.join(info['type'].split(','))),
('Weight', prop_int_pretty(info['weight'])),
('Hecke eigenform', "yes" if info['is_eigenform'] else "no"),
('Field degree', prop_int_pretty(info['fdeg']))]
try:
evs_to_show = parse_ints_to_list_flash(args.get('ev_index'),
'list of $l$')
fcs_to_show = parse_ints_to_list_flash(args.get('fc_det'),
'list of $\\det(F)$')
except ValueError:
evs_to_show = []
fcs_to_show = []
info['evs_to_show'] = sorted([
n for n in
(evs_to_show if len(evs_to_show) else sam.available_eigenvalues()[:10])
])
info['fcs_to_show'] = sorted([
n for n in (fcs_to_show if len(fcs_to_show) else sam.
available_Fourier_coefficients()[1:6])
])
info['evs_avail'] = [n for n in sam.available_eigenvalues()]
info['fcs_avail'] = [n for n in sam.available_Fourier_coefficients()]
# Do not attempt to constuct a modulus ideal unless the field has a reasonably small discriminant
# otherwise sage may not even be able to factor the discriminant
info['field'] = sam.field()
if info['field_poly'].disc() < 10**80:
null_ideal = sam.field().ring_of_integers().ideal(0)
info['modulus'] = null_ideal
modulus = args.get('modulus', '').strip()
m = 0
if modulus:
try:
O = sam.field().ring_of_integers()
m = O.ideal([O(str(b)) for b in modulus.split(',')])
except Exception:
info['error'] = True
flash_error(
"Unable to construct modulus ideal from specified generators %s.",
modulus)
if m == 1:
info['error'] = True
flash_error(
"The ideal %s is the unit ideal, please specify a different modulus.",
'(' + modulus + ')')
m = 0
info['modulus'] = m
# Hack to reduce polynomials and to handle non integral stuff
def redc(c):
return m.reduce(c * c.denominator()) / m.reduce(c.denominator())
def redp(f):
c = f.dict()
return f.parent()(dict((e, redc(c[e])) for e in c))
def safe_reduce(f):
if not m:
return latex(f)
try:
if f in sam.field():
return latex(redc(f))
else:
return latex(redp(f))
except ZeroDivisionError:
return '\\textrm{Unable to reduce} \\bmod\\mathfrak{m}'
info['reduce'] = safe_reduce
else:
info['reduce'] = latex
# check that explicit formula is not ridiculously big
if sam.explicit_formula():
info['explicit_formula_bytes'] = len(sam.explicit_formula())
if len(sam.explicit_formula()) < 100000:
info['explicit_formula'] = sam.explicit_formula()
return render_template("ModularForm_GSp4_Q_sample.html",
title=title,
bread=bread,
properties=properties,
info=info)
def render_lattice_webpage(**args):
f = None
if 'label' in args:
lab = clean_input(args.get('label'))
if lab != args.get('label'):
return redirect(url_for('.render_lattice_webpage', label=lab), 301)
f = db.lat_lattices.lucky({'$or':[{'label': lab }, {'name': {'$contains': [lab]}}]})
if f is None:
t = "Integral lattice search error"
bread = get_bread()
flash_error("%s is not a valid label or name for an integral lattice in the database.", lab)
return render_template("lattice-error.html", title=t, properties=[], bread=bread, learnmore=learnmore_list())
info = {}
info.update(f)
info['friends'] = []
bread = get_bread(f['label'])
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 dynamic 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', prop_int_pretty(info['dim'])),
('Determinant', prop_int_pretty(info['det'])),
('Level', prop_int_pretty(info['level']))]
if info['class_number'] == 0:
info['properties']=[('Class number', 'not available')]+info['properties']
else:
info['properties']=[('Class number', prop_int_pretty(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, title=t, bread=bread, properties=info['properties'], learnmore=learnmore_list(), KNOWL_ID="lattice.%s"%info['label'])
def make_form(self, nap0=50):
# To start with the data fields of self are just those from
# the database. We need to reformat these and compute some
# further (easy) data about it.
#
from lmfdb.ecnf.WebEllipticCurve import FIELD
self.field = FIELD(self.field_label)
pretty_field = field_pretty(self.field_label)
self.field_knowl = nf_display_knowl(self.field_label, pretty_field)
try:
dims = db.bmf_dims.lucky(
{
'field_label': self.field_label,
'level_label': self.level_label
},
projection='gl2_dims')
self.newspace_dimension = dims[str(self.weight)]['new_dim']
except TypeError:
self.newspace_dimension = 'not available'
self.newspace_label = "-".join([self.field_label, self.level_label])
self.newspace_url = url_for(".render_bmf_space_webpage",
field_label=self.field_label,
level_label=self.level_label)
K = self.field.K()
# 'hecke_poly_obj' is the non-LaTeX version of hecke_poly
self.hecke_poly_obj = self.hecke_poly
if self.dimension > 1:
Qx = PolynomialRing(QQ, 'x')
self.hecke_poly = Qx(str(self.hecke_poly))
F = NumberField(self.hecke_poly, 'z')
self.hecke_poly = web_latex(self.hecke_poly)
def conv(ap):
if '?' in ap:
return 'not known'
else:
return F(str(ap))
self.hecke_eigs = [conv(str(ap)) for ap in self.hecke_eigs]
self.level = ideal_from_label(K, self.level_label)
self.level_ideal2 = web_latex(self.level)
badp = self.level.prime_factors()
self.nap = len(self.hecke_eigs)
self.nap0 = min(nap0, self.nap)
self.neigs = self.nap0 + len(badp)
self.hecke_table = [[
web_latex(p.norm()),
ideal_label(p),
web_latex(p.gens_reduced()[0]),
web_latex(ap)
] for p, ap in zip(primes_iter(K), self.hecke_eigs[:self.neigs])
if not p in badp]
self.have_AL = self.AL_eigs[0] != '?'
if self.have_AL:
self.AL_table = [[
web_latex(p.norm()),
ideal_label(p),
web_latex(p.gens_reduced()[0]),
web_latex(ap)
] for p, ap in zip(badp, self.AL_eigs)]
# The following helps to create Sage download data
self.AL_table_data = [[p.gens_reduced(), ap]
for p, ap in zip(badp, self.AL_eigs)]
self.sign = 'not determined'
try:
if self.sfe == 1:
self.sign = "$+1$"
elif self.sfe == -1:
self.sign = "$-1$"
except AttributeError:
self.sfe = '?'
if self.Lratio == '?':
self.Lratio = "not determined"
self.anrank = "not determined"
else:
self.Lratio = QQ(self.Lratio)
self.anrank = r"\(0\)" if self.Lratio != 0 else "odd" if self.sfe == -1 else r"\(\ge2\), even"
self.properties = [('Label', self.label), ('Base field', pretty_field),
('Weight', prop_int_pretty(self.weight)),
('Level norm', prop_int_pretty(self.level_norm)),
('Level', self.level_ideal2),
('Dimension', prop_int_pretty(self.dimension))]
try:
if self.CM == '?':
self.CM = 'not determined'
elif self.CM == 0:
self.CM = 'no'
else:
if int(self.CM) % 4 in [2, 3]:
self.CM = 4 * int(self.CM)
self.CM = "$%s$" % self.CM
except AttributeError:
self.CM = 'not determined'
self.properties.append(('CM', str(self.CM)))
self.bc_extra = ''
self.bcd = 0
self.bct = self.bc != '?' and self.bc != 0
if self.bc == '?':
self.bc = 'not determined'
elif self.bc == 0:
self.bc = 'no'
elif self.bc == 1:
self.bcd = self.bc
self.bc = 'yes'
elif self.bc > 1:
self.bcd = self.bc
self.bc = 'yes'
self.bc_extra = r', of a form over \(\mathbb{Q}\) with coefficients in \(\mathbb{Q}(\sqrt{' + str(
self.bcd) + r'})\)'
elif self.bc == -1:
self.bc = 'no'
self.bc_extra = r', but is a twist of the base change of a form over \(\mathbb{Q}\)'
elif self.bc < -1:
self.bcd = -self.bc
self.bc = 'no'
self.bc_extra = r', but is a twist of the base change of a form over \(\mathbb{Q}\) with coefficients in \(\mathbb{Q}(\sqrt{' + str(
self.bcd) + r'})\)'
self.properties.append(('Base change', str(self.bc)))
curve_bc = db.ec_nfcurves.lucky({'class_label': self.label},
projection="base_change")
if curve_bc is not None:
if curve_bc and "." not in curve_bc[0]:
curve_bc = [
cremona_label_to_lmfdb_label(lab) for lab in curve_bc
]
self.ec_status = 'exists'
self.ec_url = url_for("ecnf.show_ecnf_isoclass",
nf=self.field_label,
conductor_label=self.level_label,
class_label=self.label_suffix)
curve_bc_parts = [split_lmfdb_label(lab) for lab in curve_bc]
bc_urls = [
url_for("cmf.by_url_newform_label",
level=cond,
weight=2,
char_orbit_label='a',
hecke_orbit=iso) for cond, iso, num in curve_bc_parts
]
bc_labels = [
".".join([str(cond), str(2), 'a', iso])
for cond, iso, _ in curve_bc_parts
]
bc_exists = [db.mf_newforms.label_exists(lab) for lab in bc_labels]
self.bc_forms = [{
'exists': ex,
'label': lab,
'url': url
} for ex, lab, url in zip(bc_exists, bc_labels, bc_urls)]
else:
self.bc_forms = []
if self.bct or self.label in bmfs_with_no_curve:
self.ec_status = 'none'
else:
self.ec_status = 'missing'
self.properties.append(('Sign', self.sign))
self.properties.append(('Analytic rank', self.anrank))
self.friends = []
self.friends += [('Newspace {}'.format(self.newspace_label),
self.newspace_url)]
url = 'ModularForm/GL2/ImaginaryQuadratic/{}'.format(
self.label.replace('-', '/'))
Lfun = get_lfunction_by_url(url)
if Lfun:
instances = get_instances_by_Lhash_and_trace_hash(
Lfun['Lhash'], Lfun['degree'], Lfun['trace_hash'])
# This will also add the EC/G2C, as this how the Lfun was computed
# and not add itself
self.friends = names_and_urls(instances, exclude={url})
self.friends.append(('L-function', '/L/' + url))
else:
# old code
if self.dimension == 1:
if self.ec_status == 'exists':
self.friends += [('Isogeny class {}'.format(self.label),
self.ec_url)]
elif self.ec_status == 'missing':
self.friends += [
('Isogeny class {} missing'.format(self.label), "")
]
else:
self.friends += [('No elliptic curve', "")]
self.friends += [('L-function not available', '')]
def make_galmap_object(self, galmap):
from lmfdb.belyi.main import url_for_belyi_passport_label
# all information about the map goes in the data dictionary
# most of the data from the database gets polished/formatted before we put it in the data dictionary
data = self.data = {}
# the stuff that does not need to be polished
for elt in ("label", "plabel", "triples_cyc", "orbit_size", "g", "abc",
"deg"):
data[elt] = galmap[elt]
nt = galmap["group"].split("T")
data["group"] = group_display_knowl(int(nt[0]), int(nt[1]))
data["geomtype"] = geomtypelet_to_geomtypename_dict[galmap["geomtype"]]
data["lambdas"] = [str(c)[1:-1] for c in galmap["lambdas"]]
data["isQQ"] = False
data["in_LMFDB"] = False
F = belyi_base_field(galmap)
if F._data is None:
fld_coeffs = galmap["base_field"]
pol = PolynomialRing(QQ, "x")(fld_coeffs)
data["base_field"] = latex(pol)
else:
data["in_LMFDB"] = True
if F.poly().degree() == 1:
data["isQQ"] = True
F.latex_poly = web_latex(F.poly())
data["base_field"] = F
crv_str = galmap["curve"]
if crv_str == "PP1":
data["curve"] = r"\mathbb{P}^1"
else:
data["curve"] = make_curve_latex(crv_str)
#if galmap['curve_label']:
# data['curve_label'] = galmap['curve_label']
# change pairs of floats to complex numbers
embeds = galmap["embeddings"]
embed_strs = []
for el in embeds:
if el[1] < 0:
el_str = str(el[0]) + str(el[1]) + r"\sqrt{-1}"
else:
el_str = str(el[0]) + "+" + str(el[1]) + r"\sqrt{-1}"
embed_strs.append(el_str)
data["map"] = make_map_latex(galmap["map"])
data["embeddings_and_triples"] = []
if data["isQQ"]:
for i in range(0, len(data["triples_cyc"])):
triple_cyc = data["triples_cyc"][i]
data["embeddings_and_triples"].append([
r"\text{not applicable (over $\mathbb{Q}$)}",
triple_cyc[0],
triple_cyc[1],
triple_cyc[2],
])
else:
for i in range(0, len(data["triples_cyc"])):
triple_cyc = data["triples_cyc"][i]
data["embeddings_and_triples"].append([
embed_strs[i], triple_cyc[0], triple_cyc[1], triple_cyc[2]
])
data["lambdas"] = [str(c)[1:-1] for c in galmap["lambdas"]]
# Properties
self.plot = db.belyi_galmap_portraits.lucky({"label": galmap['label']},
projection="portrait")
plot_link = '<a href="{0}"><img src="{0}" width="200" height="200" style="background-color: white;"/></a>'.format(
self.plot)
properties = [("Label", galmap["label"])]
if self.plot:
properties += [(None, plot_link)]
properties += [
("Group", str(galmap["group"])),
("Orders", "$%s$" % (galmap["abc"])),
("Genus", prop_int_pretty(galmap["g"])),
("Size", prop_int_pretty(galmap["orbit_size"])),
]
self.properties = properties
# Friends
self.friends = [("Passport",
url_for_belyi_passport_label(galmap["plabel"]))]
self.friends.extend(names_and_urls(galmap['friends']))
# Downloads
if galmap["g"] <= 2:
self.downloads = [
(
"Code to Magma",
url_for(".belyi_galmap_magma_download",
label=data["label"]),
),
(
"Code to SageMath",
url_for(".belyi_galmap_sage_download",
label=data["label"]),
),
(
"All data to text",
url_for(".belyi_galmap_text_download",
label=data["label"]),
),
]
else:
self.downloads = []
# Breadcrumbs
label_spl = data["label"].split("-")
groupstr = label_spl[0]
letnum = label_spl[2]
gstr = str(data['g'])
sigmas = label_spl[1]
sigma0, sigma1, sigmaoo = sigmas.split("_")
abcstr = str(data['abc']).replace(' ', '')
# does lambdasstr need to be updated?
lambdasstr = "%s-%s-%s" % (sigma0, sigma1, sigmaoo)
lambdasgstr = lambdasstr + "-" + gstr
self.bread = [
("Belyi Maps", url_for(".index")),
(groupstr, url_for(".by_url_belyi_search_group", group=groupstr)),
(
abcstr,
url_for(".by_url_belyi_search_group_triple",
group=groupstr,
abc=abcstr),
),
(
lambdasgstr,
url_for(
".by_url_belyi_passport_label",
group=groupstr,
abc=abcstr,
sigma0=sigma0,
sigma1=sigma1,
sigmaoo=sigmaoo,
g=gstr,
),
),
(
letnum,
url_for(
".by_url_belyi_galmap_label",
group=groupstr,
abc=abcstr,
sigma0=sigma0,
sigma1=sigma1,
sigmaoo=sigmaoo,
g=gstr,
letnum=letnum,
),
),
]
# Title
self.title = "Belyi map orbit " + data["label"]
# Code snippets (only for curves)
self.code = {}
return
def render_group_webpage(args):
data = {}
if 'label' in args:
label = clean_input(args['label'])
label = label.replace('t', 'T')
data = db.gps_transitive.lookup(label)
if data is None:
if re.match(r'^\d+T\d+$', label):
flash_error("Group %s was not found in the database.", label)
else:
flash_error("%s is not a valid label for a Galois group.",
label)
return redirect(url_for(".index"))
title = 'Galois group: ' + label
wgg = WebGaloisGroup.from_nt(data['n'], data['t'])
data['wgg'] = wgg
n = data['n']
t = data['t']
data['yesno'] = yesno
order = data['order']
data['orderfac'] = latex(ZZ(order).factor())
orderfac = latex(ZZ(order).factor())
data['ordermsg'] = "$%s=%s$" % (order, latex(orderfac))
if order == 1:
data['ordermsg'] = "$1$"
if ZZ(order).is_prime():
data['ordermsg'] = "$%s$ (is prime)" % order
pgroup = len(ZZ(order).prime_factors()) < 2
if wgg.num_conjclasses() < 50:
data['cclasses'] = wgg.conjclasses()
if ZZ(order) < ZZ(10000000) and wgg.num_conjclasses() < 21:
data['chartable'] = chartable(n, t)
data['gens'] = wgg.generator_string()
if n == 1 and t == 1:
data['gens'] = 'None needed'
data['num_cc'] = comma(wgg.num_conjclasses())
data['parity'] = "$%s$" % data['parity']
data['subinfo'] = subfield_display(n, data['subfields'])
data['resolve'] = resolve_display(data['quotients'])
if data['gapid'] == 0:
data['gapid'] = "not available"
else:
data['gapid'] = small_group_display_knowl(
int(data['order']), int(data['gapid']),
str([int(data['order']),
int(data['gapid'])]))
data['otherreps'] = wgg.otherrep_list()
ae = data['arith_equiv']
if ae > 0:
if ae > 1:
data[
'arith_equiv'] = r'A number field with this Galois group has %d <a knowl="nf.arithmetically_equivalent", title="arithmetically equivalent">arithmetically equivalent</a> fields.' % ae
else:
data[
'arith_equiv'] = r'A number field with this Galois group has exactly one <a knowl="nf.arithmetically_equivalent", title="arithmetically equivalent">arithmetically equivalent</a> field.'
elif ae > -1:
data[
'arith_equiv'] = r'A number field with this Galois group has no <a knowl="nf.arithmetically_equivalent", title="arithmetically equivalent">arithmetically equivalent</a> fields.'
else:
data[
'arith_equiv'] = r'Data on whether or not a number field with this Galois group has <a knowl="nf.arithmetically_equivalent", title="arithmetically equivalent">arithmetically equivalent</a> fields has not been computed.'
intreps = list(db.gps_gmodules.search({'n': n, 't': t}))
if len(intreps) > 0:
data['int_rep_classes'] = [str(z[0]) for z in intreps[0]['gens']]
for onerep in intreps:
onerep['gens'] = [
list_to_latex_matrix(z[1]) for z in onerep['gens']
]
data['int_reps'] = intreps
data['int_reps_complete'] = int_reps_are_complete(intreps)
dcq = data['moddecompuniq']
if dcq[0] == 0:
data['decompunique'] = 0
else:
data['decompunique'] = dcq[0]
data['isoms'] = [[mult2mult(z[0]),
mult2mult(z[1])] for z in dcq[1]]
data['isoms'] = [[
modules2string(n, t, z[0]),
modules2string(n, t, z[1])
] for z in data['isoms']]
#print dcq[1]
#print data['isoms']
friends = []
if db.nf_fields.exists({'galois_label': "%dT%d" % (n, t)}):
friends.append(
('Number fields with this Galois group',
url_for('number_fields.number_field_render_webpage') +
"?galois_group=%dT%d" % (n, t)))
prop2 = [
('Label', label),
('Degree', prop_int_pretty(data['n'])),
('Order', prop_int_pretty(order)),
('Cyclic', yesno(data['cyc'])),
('Abelian', yesno(data['ab'])),
('Solvable', yesno(data['solv'])),
('Primitive', yesno(data['prim'])),
('$p$-group', yesno(pgroup)),
]
pretty = group_display_short(n, t, emptyifnotpretty=True)
if len(pretty) > 0:
prop2.extend([('Group:', pretty)])
data['pretty_name'] = pretty
data['name'] = re.sub(r'_(\d+)', r'_{\1}', data['name'])
data['name'] = re.sub(r'\^(\d+)', r'^{\1}', data['name'])
data['nilpotency'] = '$%s$' % data['nilpotency']
if data['nilpotency'] == '$-1$':
data['nilpotency'] += ' (not nilpotent)'
bread = get_bread([(label, ' ')])
return render_template("gg-show-group.html",
credit=GG_credit,
title=title,
bread=bread,
info=data,
properties=prop2,
friends=friends,
KNOWL_ID="gg.%s" % label,
learnmore=learnmore_list())
def make_curve(self):
# To start with the data fields of self are just those from
# the database. We need to reformat these.
# Old version: required constructing the actual elliptic curve
# E, and computing some further data about it.
# New version (May 2016): extra data fields now in the
# database so we do not have to construct the curve or do any
# computation with it on the fly. As a failsafe the old way
# is still included.
data = self.data = {}
data['ainvs'] = [ZZ(ai) for ai in self.ainvs]
data['conductor'] = N = ZZ(self.conductor)
data['j_invariant'] = QQ(str(self.jinv))
data['j_inv_factor'] = latex(0)
if data['j_invariant']: # don't factor 0
data['j_inv_factor'] = latex(data['j_invariant'].factor())
data['j_inv_latex'] = web_latex(data['j_invariant'])
# extract data about MW rank, generators, heights and torsion:
self.make_mw()
# get more data from the database entry
data['equation'] = self.equation
local_data = self.local_data
D = self.signD * prod([ld['p']**ld['ord_disc'] for ld in local_data])
for ld in local_data:
ld['kod'] = ld['kod'].replace("\\\\", "\\")
data['disc'] = D
Nfac = Factorization([(ZZ(ld['p']), ld['ord_cond'])
for ld in local_data])
Dfac = Factorization([(ZZ(ld['p']), ld['ord_disc'])
for ld in local_data],
unit=ZZ(self.signD))
data['minq_D'] = minqD = self.min_quad_twist['disc']
data['minq_label'] = self.min_quad_twist[
'lmfdb_label'] if self.label_type == 'LMFDB' else self.min_quad_twist[
'label']
data['minq_info'] = '(itself)' if minqD == 1 else '(by {})'.format(
minqD)
if self.degree is None:
data['degree'] = 0 # invalid, but will be displayed nicely
else:
data['degree'] = self.degree
try:
data['an'] = self.anlist
data['ap'] = self.aplist
except AttributeError:
r = db.ec_curves.lucky({'lmfdb_iso': self.lmfdb_iso, 'number': 1})
data['an'] = r['anlist']
data['ap'] = r['aplist']
data['disc_factor'] = latex(Dfac)
data['cond_factor'] = latex(Nfac)
data['disc_latex'] = web_latex(D)
data['cond_latex'] = web_latex(N)
data['galois_images'] = [
trim_galois_image_code(s) for s in self.mod_p_images
]
data['non_maximal_primes'] = self.non_maximal_primes
data['galois_data'] = [{
'p': p,
'image': im
} for p, im in zip(data['non_maximal_primes'], data['galois_images'])]
data['CMD'] = self.cm
data['CM'] = "no"
data['EndE'] = r"\(\Z\)"
if self.cm:
data['cm_ramp'] = [
p for p in ZZ(self.cm).support()
if not p in self.non_maximal_primes
]
data['cm_nramp'] = len(data['cm_ramp'])
if data['cm_nramp'] == 1:
data['cm_ramp'] = data['cm_ramp'][0]
else:
data['cm_ramp'] = ", ".join([str(p) for p in data['cm_ramp']])
data['cm_sqf'] = ZZ(self.cm).squarefree_part()
data['CM'] = r"yes (\(D=%s\))" % data['CMD']
if data['CMD'] % 4 == 0:
d4 = ZZ(data['CMD']) // 4
data['EndE'] = r"\(\Z[\sqrt{%s}]\)" % d4
else:
data['EndE'] = r"\(\Z[(1+\sqrt{%s})/2]\)" % data['CMD']
data['ST'] = st_link_by_name(1, 2, 'N(U(1))')
else:
data['ST'] = st_link_by_name(1, 2, 'SU(2)')
data['p_adic_primes'] = [
p for i, p in enumerate(prime_range(5, 100))
if (N * data['ap'][i]) % p != 0
]
cond, iso, num = split_lmfdb_label(self.lmfdb_label)
self.one_deg = ZZ(self.class_deg).is_prime()
isodegs = [str(d) for d in self.isogeny_degrees if d > 1]
if len(isodegs) < 3:
data['isogeny_degrees'] = " and ".join(isodegs)
else:
data['isogeny_degrees'] = " and ".join(
[", ".join(isodegs[:-1]), isodegs[-1]])
if self.twoadic_gens:
from sage.matrix.all import Matrix
data['twoadic_gen_matrices'] = ','.join(
[latex(Matrix(2, 2, M)) for M in self.twoadic_gens])
data[
'twoadic_rouse_url'] = ROUSE_URL_PREFIX + self.twoadic_label + ".html"
# Leading term of L-function & other BSD data
self.make_bsd()
# Optimality (the optimal curve in the class is the curve
# whose Cremona label ends in '1' except for '990h' which was
# labelled wrongly long ago): this is proved for N up to
# OPTIMALITY_BOUND (and when there is only one curve in an
# isogeny class, obviously) and expected for all N.
# Column 'optimality' is 1 for certainly optimal curves, 0 for
# certainly non-optimal curves, and is n>1 if the curve is one
# of n in the isogeny class which may be optimal given current
# knowledge.
# Column "manin_constant' is the correct Manin constant
# assuming that the optimal curve in the class is known, or
# otherwise if it is the curve with (Cremona) number 1.
# The code here allows us to update the display correctly by
# changing one line in this file (defining OPTIMALITY_BOUND)
# without changing the data.
data['optimality_bound'] = OPTIMALITY_BOUND
data[
'manin_constant'] = self.manin_constant # (conditional on data['optimality_known'])
if N < OPTIMALITY_BOUND:
data['optimality_code'] = int(
self.number == (3 if self.iso == '990h' else 1))
data['optimality_known'] = True
data['manin_known'] = True
if self.label_type == 'Cremona':
data[
'optimal_label'] = '990h3' if self.iso == '990h' else self.iso + '1'
else:
data[
'optimal_label'] = '990.i3' if self.lmfdb_iso == '990.i' else self.lmfdb_iso + '1'
else:
data['optimality_code'] = self.optimality
data['optimality_known'] = (self.optimality < 2)
if self.optimality == 1:
data['manin_known'] = True
data[
'optimal_label'] = self.label if self.label_type == 'Cremona' else self.lmfdb_label
else:
if self.number == 1:
data['manin_known'] = False
data[
'optimal_label'] = self.label if self.label_type == 'Cremona' else self.lmfdb_label
else:
# find curve #1 in this class and its optimailty code:
opt_curve = db.ec_curves.lucky(
{
'iso': self.iso,
'number': 1
},
projection=['label', 'lmfdb_label', 'optimality'])
data['manin_known'] = (opt_curve['optimality'] == 1)
data['optimal_label'] = opt_curve[
'label' if self.label_type ==
'Cremona' else 'lmfdb_label']
data['p_adic_data_exists'] = False
if data['optimality_code'] == 1:
data['p_adic_data_exists'] = db.ec_padic.exists(
{'lmfdb_iso': self.lmfdb_iso})
# Iwasawa data (where present)
self.make_iwasawa()
# Torsion growth data (where present)
self.make_torsion_growth()
data['newform'] = web_latex(
PowerSeriesRing(QQ, 'q')(data['an'], 20, check=True))
data['newform_label'] = self.newform_label = ".".join(
[str(cond), str(2), 'a', iso])
self.newform_link = url_for("cmf.by_url_newform_label",
level=cond,
weight=2,
char_orbit_label='a',
hecke_orbit=iso)
self.newform_exists_in_db = db.mf_newforms.label_exists(
self.newform_label)
self._code = None
if self.label_type == 'Cremona':
self.class_url = url_for(".by_ec_label", label=self.iso)
self.class_name = self.iso
else:
self.class_url = url_for(".by_double_iso_label",
conductor=N,
iso_label=iso)
self.class_name = self.lmfdb_iso
data['class_name'] = self.class_name
data['number'] = self.number
self.friends = [('Isogeny class ' + self.class_name, self.class_url),
('Minimal quadratic twist %s %s' %
(data['minq_info'], data['minq_label']),
url_for(".by_ec_label", label=data['minq_label'])),
('All twists ',
url_for(".rational_elliptic_curves", jinv=self.jinv))]
lfun_url = url_for("l_functions.l_function_ec_page",
conductor_label=N,
isogeny_class_label=iso)
origin_url = lfun_url.lstrip('/L/').rstrip('/')
if db.lfunc_instances.exists({'url': origin_url}):
self.friends += [('L-function', lfun_url)]
else:
self.friends += [('L-function not available', "")]
if not self.cm:
if N <= 300:
self.friends += [('Symmetric square L-function',
url_for("l_functions.l_function_ec_sym_page",
power='2',
conductor=N,
isogeny=iso))]
if N <= 50:
self.friends += [('Symmetric cube L-function',
url_for("l_functions.l_function_ec_sym_page",
power='3',
conductor=N,
isogeny=iso))]
if self.newform_exists_in_db:
self.friends += [('Modular form ' + self.newform_label,
self.newform_link)]
self.downloads = [('q-expansion to text',
url_for(".download_EC_qexp",
label=self.lmfdb_label,
limit=1000)),
('All stored data to text',
url_for(".download_EC_all",
label=self.lmfdb_label)),
('Code to Magma',
url_for(".ec_code_download",
conductor=cond,
iso=iso,
number=num,
label=self.lmfdb_label,
download_type='magma')),
('Code to SageMath',
url_for(".ec_code_download",
conductor=cond,
iso=iso,
number=num,
label=self.lmfdb_label,
download_type='sage')),
('Code to GP',
url_for(".ec_code_download",
conductor=cond,
iso=iso,
number=num,
label=self.lmfdb_label,
download_type='gp'))]
try:
self.plot = encode_plot(self.E.plot())
except AttributeError:
self.plot = encode_plot(EllipticCurve(data['ainvs']).plot())
self.plot_link = '<a href="{0}"><img src="{0}" width="200" height="150"/></a>'.format(
self.plot)
self.properties = [
('Label',
self.label if self.label_type == 'Cremona' else self.lmfdb_label),
(None, self.plot_link),
('Conductor', prop_int_pretty(data['conductor'])),
('Discriminant', prop_int_pretty(data['disc'])),
('j-invariant', '%s' % data['j_inv_latex']),
('CM', '%s' % data['CM']),
('Rank', prop_int_pretty(self.mw['rank'])),
('Torsion structure',
(r'\(%s\)' %
self.mw['tor_struct']) if self.mw['tor_struct'] else 'trivial'),
]
if self.label_type == 'Cremona':
self.title = "Elliptic curve with Cremona label {} (LMFDB label {})".format(
self.label, self.lmfdb_label)
else:
self.title = "Elliptic curve with LMFDB label {} (Cremona label {})".format(
self.lmfdb_label, self.label)
self.bread = [('Elliptic curves', url_for("ecnf.index")),
(r'$\Q$', url_for(".rational_elliptic_curves")),
('%s' % N, url_for(".by_conductor", conductor=N)),
('%s' % iso,
url_for(".by_double_iso_label",
conductor=N,
iso_label=iso)), ('%s' % num, ' ')]
def __init__(self, data):
# Need to set mf_dim, eis_dim, cusp_dim, new_dim, old_dim
self.__dict__.update(data)
self.factored_level = web_latex_factored_integer(self.level,
equals=True)
self.has_projective_image_types = all(typ + '_dim' in data
for typ in ('dihedral', 'a4',
's4', 'a5'))
# The following can be removed once we change the behavior of lucky to include Nones
self.num_forms = data.get('num_forms')
self.trace_bound = data.get('trace_bound')
self.has_trace_form = (data.get('traces') is not None)
self.char_conrey = self.conrey_indexes[0]
self.char_conrey_str = r'\chi_{%s}(%s,\cdot)' % (self.level,
self.char_conrey)
self.newforms = list(
db.mf_newforms.search({'space_label': self.label}, projection=2))
oldspaces = db.mf_subspaces.search(
{
'label': self.label,
'sub_level': {
'$ne': self.level
}
}, [
'sub_level', 'sub_char_orbit_index', 'sub_conrey_indexes',
'sub_mult'
])
self.oldspaces = [(old['sub_level'], old['sub_char_orbit_index'],
old['sub_conrey_indexes'][0], old['sub_mult'])
for old in oldspaces]
self.dim_grid = DimGrid.from_db(data)
self.plot = db.mf_newspace_portraits.lookup(self.label,
projection="portrait")
# Properties
self.properties = [('Label', self.label)]
if self.plot is not None and self.dim > 0:
self.properties += [
(None,
'<img src="{0}" width="200" height="200"/>'.format(self.plot))
]
self.properties += [
('Level', prop_int_pretty(self.level)),
('Weight', prop_int_pretty(self.weight)),
('Character orbit', '%s.%s' % (self.level, self.char_orbit_label)),
('Rep. character', '$%s$' % self.char_conrey_str),
('Character field',
r'$\Q%s$' % ('' if self.char_degree == 1 else r'(\zeta_{%s})' %
self.char_order)),
('Dimension', prop_int_pretty(self.dim)),
]
if self.num_forms is not None:
self.properties.append(
('Newform subspaces', prop_int_pretty(self.num_forms)))
self.properties.append(
('Sturm bound', prop_int_pretty(self.sturm_bound)))
if data.get('trace_bound') is not None:
self.properties.append(
('Trace bound', prop_int_pretty(self.trace_bound)))
# Work around search results not including None
if data.get('num_forms') is None:
self.num_forms = None
# Breadcrumbs
self.bread = get_bread(level=self.level,
weight=self.weight,
char_orbit_label=self.char_orbit_label)
# Downloads
self.downloads = [
('Trace form to text',
url_for('cmf.download_traces', label=self.label)),
('All stored data to text',
url_for('.download_newspace', label=self.label)),
]
if self.conrey_indexes[0] == 1:
self.trivial_character = True
character_str = "trivial character"
if self.dim == 0:
self.dim_str = r"\(%s\)" % (self.dim)
else:
self.minus_dim = self.dim - self.plus_dim
self.dim_str = r"\(%s + %s\)" % (self.plus_dim, self.minus_dim)
else:
self.trivial_character = False
character_str = r"Character {level}.{orbit_label}".format(
level=self.level, orbit_label=self.char_orbit_label)
# character_str = r"Character \(\chi_{{{level}}}({conrey}, \cdot)\)".format(level=self.level, conrey=self.conrey_indexes[0])
self.dim_str = r"\(%s\)" % (self.dim)
self.title = r"Space of modular forms of level %s, weight %s, and %s" % (
self.level, self.weight, character_str)
gamma1_link = '/ModularForm/GL2/Q/holomorphic/%d/%d' % (self.level,
self.weight)
self.friends = [('Newspace %d.%d' % (self.level, self.weight),
gamma1_link)]
def __init__(self, galmap, triple=None):
from lmfdb.belyi.main import url_for_belyi_passport_label, url_for_belyi_galmap_label
# all information about the map goes in the data dictionary
# most of the data from the database gets polished/formatted before we put it in the data dictionary
data = self.data = {}
# the stuff that does not need to be polished
for elt in ("label", "plabel", "triples_cyc", "orbit_size", "g", "abc",
"deg", "primitivization", "is_primitive"):
data[elt] = galmap[elt]
if triple:
data["label"] += '-' + (triple).replace(' ', '')
data["triple"] = triple
data["group"] = transitive_group_display_knowl(galmap["group"])
data["geomtype"] = geomtypelet_to_geomtypename_dict[galmap["geomtype"]]
data["lambdas"] = [str(c)[1:-1] for c in galmap["lambdas"]]
data["primitivization_url"] = url_for_belyi_galmap_label(
data['primitivization'])
data["isQQ"] = False
data["in_LMFDB"] = False
F = belyi_base_field(galmap)
if F._data is None:
fld_coeffs = galmap["base_field"]
pol = PolynomialRing(QQ, "t")(fld_coeffs)
data["base_field"] = latex(pol)
else:
data["in_LMFDB"] = True
if F.poly().degree() == 1:
data["isQQ"] = True
F.latex_poly = web_latex(F.poly(var="t"))
data["base_field"] = F
data['embeddings'] = galmap['embeddings']
# change pairs of floats to complex numbers
embed_strs = []
for el in galmap["embeddings"]:
if el[1] < 0:
el_str = str(el[0]) + str(el[1]) + r"\sqrt{-1}"
else:
el_str = str(el[0]) + "+" + str(el[1]) + r"\sqrt{-1}"
embed_strs.append(el_str)
data["embeddings_and_triples"] = []
self.triple = None
self.embedding = None
for i in range(0, len(data["triples_cyc"])):
my_dict = {}
triple_str = ', '.join(data['triples_cyc'][i])
triple_link = triple_str.replace(' ', '')
if triple_link == triple:
self.triple = data['triples_cyc'][i]
self.embedding = CC(data['embeddings'][i])
my_dict['triple'] = triple_str
my_dict['triple_link'] = triple_link
if data["isQQ"]:
my_dict[
'embedding'] = r"\text{not applicable (over $\mathbb{Q}$)}"
else:
my_dict['embedding'] = embed_strs[i]
data['embeddings_and_triples'].append(my_dict)
crv_str = galmap["curve"]
if crv_str == "PP1":
data["curve"] = r"\mathbb{P}^1"
else:
data["curve"] = make_curve_latex(crv_str, nu=self.embedding)
data["map"] = make_map_latex(galmap["map"], nu=self.embedding)
data["lambdas"] = [str(c)[1:-1] for c in galmap["lambdas"]]
# Properties
self.plot = db.belyi_galmap_portraits.lucky({"label": galmap['label']},
projection="portrait")
plot_link = '<a href="{0}"><img src="{0}" width="200" height="200" style="background-color: white;"/></a>'.format(
self.plot)
properties = [("Label", galmap["label"])]
if triple:
properties += [("Triple", "$%s$" % triple)]
if self.plot:
properties += [(None, plot_link)]
properties += [
("Group", str(galmap["group"])),
("Orders", "$%s$" % (data["abc"])),
("Genus", prop_int_pretty(data["g"])),
("Size", prop_int_pretty(data["orbit_size"])),
]
self.properties = properties
# Friends
self.friends = [("Passport",
url_for_belyi_passport_label(galmap["plabel"]))]
if galmap['label'] != galmap['primitivization']:
self.friends.append(
("Primitivization",
url_for_belyi_galmap_label(galmap["primitivization"])))
self.friends.extend(names_and_urls(galmap['friends']))
#add curve link, if in LMFDB
if 'curve_label' in galmap.keys():
data['curve_label'] = galmap['curve_label']
for name, url in self.friends:
if "curve" in name.lower() and data['curve_label'] in name:
data["curve_url"] = url
# Downloads
if galmap["g"] <= 2:
data_label = data["label"]
if triple:
spl = data_label.split("-")
data_label = "-".join(spl[0:-1])
self.downloads = [
(
"Code to Magma",
url_for(".belyi_galmap_magma_download", label=data_label),
),
(
"Code to SageMath",
url_for(".belyi_galmap_sage_download", label=data_label),
),
(
"All data to text",
url_for(".belyi_galmap_text_download", label=data_label),
),
]
else:
self.downloads = []
# Breadcrumbs
label_spl = data["label"].split("-")
groupstr = label_spl[0]
letnum = label_spl[2]
gstr = str(data['g'])
sigmas = label_spl[1]
sigma0, sigma1, sigmaoo = sigmas.split("_")
abcstr = str(data['abc']).replace(' ', '')
# does lambdasstr need to be updated?
lambdasstr = "%s-%s-%s" % (sigma0, sigma1, sigmaoo)
lambdasgstr = lambdasstr + "-" + gstr
self.bread = [
("Belyi Maps", url_for(".index")),
(groupstr, url_for(".by_url_belyi_search_group", group=groupstr)),
(
abcstr,
url_for(".by_url_belyi_search_group_triple",
group=groupstr,
abc=abcstr),
),
(
lambdasgstr,
url_for(
".by_url_belyi_passport_label",
group=groupstr,
abc=abcstr,
sigma0=sigma0,
sigma1=sigma1,
sigmaoo=sigmaoo,
g=gstr,
),
),
(
letnum,
url_for(
".by_url_belyi_galmap_label",
group=groupstr,
abc=abcstr,
sigma0=sigma0,
sigma1=sigma1,
sigmaoo=sigmaoo,
g=gstr,
letnum=letnum,
),
),
]
# Title
if self.triple:
self.title = "Embedded Belyi map " + data["label"]
else:
self.title = "Belyi map orbit " + data["label"]
# Code snippets (only for curves)
self.code = {}
self.__dict__.update(data)
return
def make_object(self, curve, endo, tama, ratpts, clus, is_curve):
from lmfdb.genus2_curves.main import url_for_curve_label
# all information about the curve, its Jacobian, isogeny class, and endomorphisms goes in the data dictionary
# most of the data from the database gets polished/formatted before we put it in the data dictionary
data = self.data = {}
data['label'] = curve['label'] if is_curve else curve['class']
data['slabel'] = data['label'].split('.')
# set attributes common to curves and isogeny classes here
data['Lhash'] = str(curve['Lhash'])
data['cond'] = ZZ(curve['cond'])
data['cond_factor_latex'] = web_latex_factored_integer(data['cond'])
data['analytic_rank'] = ZZ(curve['analytic_rank'])
data['mw_rank'] = ZZ(0) if curve.get('mw_rank') is None else ZZ(curve['mw_rank']) # 0 will be marked as a lower bound
data['mw_rank_proved'] = curve['mw_rank_proved']
data['analytic_rank_proved'] = curve['analytic_rank_proved']
data['hasse_weil_proved'] = curve['hasse_weil_proved']
data['st_group'] = curve['st_group']
data['st_group_link'] = st_link_by_name(1,4,data['st_group'])
data['st0_group_name'] = st0_group_name(curve['real_geom_end_alg'])
data['is_gl2_type'] = curve['is_gl2_type']
data['root_number'] = ZZ(curve['root_number'])
data['lfunc_url'] = url_for("l_functions.l_function_genus2_page", cond=data['slabel'][0], x=data['slabel'][1])
data['bad_lfactors'] = literal_eval(curve['bad_lfactors'])
data['bad_lfactors_pretty'] = [ (c[0], list_to_factored_poly_otherorder(c[1])) for c in data['bad_lfactors']]
if is_curve:
# invariants specific to curve
data['class'] = curve['class']
data['abs_disc'] = ZZ(curve['abs_disc'])
data['disc'] = curve['disc_sign'] * data['abs_disc']
data['min_eqn'] = literal_eval(curve['eqn'])
data['min_eqn_display'] = min_eqns_pretty(data['min_eqn'])
data['disc_factor_latex'] = web_latex_factored_integer(data['disc'])
data['igusa_clebsch'] = [ZZ(a) for a in literal_eval(curve['igusa_clebsch_inv'])]
data['igusa'] = [ZZ(a) for a in literal_eval(curve['igusa_inv'])]
data['g2'] = [QQ(a) for a in literal_eval(curve['g2_inv'])]
data['igusa_clebsch_factor_latex'] = [web_latex_factored_integer(i) for i in data['igusa_clebsch']]
data['igusa_factor_latex'] = [ web_latex_factored_integer(j) for j in data['igusa'] ]
data['aut_grp'] = small_group_label_display_knowl('%d.%d' % tuple(literal_eval(curve['aut_grp_id'])))
data['geom_aut_grp'] = small_group_label_display_knowl('%d.%d' % tuple(literal_eval(curve['geom_aut_grp_id'])))
data['num_rat_wpts'] = ZZ(curve['num_rat_wpts'])
data['has_square_sha'] = "square" if curve['has_square_sha'] else "twice a square"
P = curve['non_solvable_places']
if len(P):
sz = "except over "
sz += ", ".join([QpName(p) for p in P])
last = " and"
if len(P) > 2:
last = ", and"
sz = last.join(sz.rsplit(",",1))
else:
sz = "everywhere"
data['non_solvable_places'] = sz
data['two_selmer_rank'] = ZZ(curve['two_selmer_rank'])
data['torsion_order'] = curve['torsion_order']
data['end_ring_base'] = endo['ring_base']
data['end_ring_geom'] = endo['ring_geom']
data['real_period'] = decimal_pretty(str(curve['real_period']))
data['regulator'] = decimal_pretty(str(curve['regulator'])) if curve['regulator'] > -0.5 else 'unknown'
if data['mw_rank'] == 0 and data['mw_rank_proved']:
data['regulator'] = '1' # display an exact 1 when we know this
data['tamagawa_product'] = ZZ(curve['tamagawa_product']) if curve.get('tamagawa_product') else 0
data['analytic_sha'] = ZZ(curve['analytic_sha']) if curve.get('analytic_sha') else 0
data['leading_coeff'] = decimal_pretty(str(curve['leading_coeff'])) if curve['leading_coeff'] else 'unknown'
data['rat_pts'] = ratpts['rat_pts']
data['rat_pts_v'] = ratpts['rat_pts_v']
data['rat_pts_table'] = ratpts_table(ratpts['rat_pts'],ratpts['rat_pts_v'])
data['rat_pts_simple_table'] = ratpts_simpletable(ratpts['rat_pts'],ratpts['rat_pts_v'],data['min_eqn'])
data['mw_gens_v'] = ratpts['mw_gens_v']
lower = len([n for n in ratpts['mw_invs'] if n == 0])
upper = data['analytic_rank']
invs = ratpts['mw_invs'] if data['mw_gens_v'] or lower >= upper else [0 for n in range(upper-lower)] + ratpts['mw_invs']
if len(invs) == 0:
data['mw_group'] = 'trivial'
else:
data['mw_group'] = r'\(' + r' \times '.join([ (r'\Z' if n == 0 else r'\Z/{%s}\Z' % n) for n in invs]) + r'\)'
if lower >= upper:
data['mw_gens_table'] = mw_gens_table (ratpts['mw_invs'], ratpts['mw_gens'], ratpts['mw_heights'], ratpts['rat_pts'])
data['mw_gens_simple_table'] = mw_gens_simple_table (ratpts['mw_invs'], ratpts['mw_gens'], ratpts['mw_heights'], ratpts['rat_pts'], data['min_eqn'])
if curve['two_torsion_field'][0]:
data['two_torsion_field_knowl'] = nf_display_knowl (curve['two_torsion_field'][0], field_pretty(curve['two_torsion_field'][0]))
else:
t = curve['two_torsion_field']
data['two_torsion_field_knowl'] = r"splitting field of \(%s\) with Galois group %s" % (intlist_to_poly(t[1]),group_display_knowl(t[2][0],t[2][1]))
tamalist = [[item['p'],item['tamagawa_number']] for item in tama]
data['local_table'] = local_table (data['cond'],data['abs_disc'],tamalist,data['bad_lfactors_pretty'],clus)
else:
# invariants specific to isogeny class
curves_data = list(db.g2c_curves.search({"class" : curve['class']}, ['label','eqn']))
if not curves_data:
raise KeyError("No curves found in database for isogeny class %s of genus 2 curve %s." %(curve['class'],curve['label']))
data['curves'] = [ {"label" : c['label'], "equation_formatted" : min_eqn_pretty(literal_eval(c['eqn'])), "url": url_for_curve_label(c['label'])} for c in curves_data ]
lfunc_data = db.lfunc_lfunctions.lucky({'Lhash':str(curve['Lhash'])})
if not lfunc_data:
raise KeyError("No Lfunction found in database for isogeny class of genus 2 curve %s." %curve['label'])
if lfunc_data and lfunc_data.get('euler_factors'):
data['good_lfactors'] = [[nth_prime(n+1),lfunc_data['euler_factors'][n]] for n in range(len(lfunc_data['euler_factors'])) if nth_prime(n+1) < 30 and (data['cond'] % nth_prime(n+1))]
data['good_lfactors_pretty'] = [ (c[0], list_to_factored_poly_otherorder(c[1])) for c in data['good_lfactors']]
# Endomorphism data over QQ:
data['gl2_statement_base'] = gl2_statement_base(endo['factorsRR_base'], r'\(\Q\)')
data['factorsQQ_base'] = endo['factorsQQ_base']
data['factorsRR_base'] = endo['factorsRR_base']
data['end_statement_base'] = (r"Endomorphism %s over \(\Q\):<br>" %("ring" if is_curve else "algebra") +
end_statement(data['factorsQQ_base'], endo['factorsRR_base'], ring=data['end_ring_base'] if is_curve else None))
# Field over which all endomorphisms are defined
data['end_field_label'] = endo['fod_label']
data['end_field_poly'] = intlist_to_poly(endo['fod_coeffs'])
data['end_field_statement'] = end_field_statement(data['end_field_label'], data['end_field_poly'])
# Endomorphism data over QQbar:
data['factorsQQ_geom'] = endo['factorsQQ_geom']
data['factorsRR_geom'] = endo['factorsRR_geom']
if data['end_field_label'] != '1.1.1.1':
data['gl2_statement_geom'] = gl2_statement_base(data['factorsRR_geom'], r'\(\overline{\Q}\)')
data['end_statement_geom'] = (r"Endomorphism %s over \(\overline{\Q}\):" %("ring" if is_curve else "algebra") +
end_statement(data['factorsQQ_geom'], data['factorsRR_geom'], field=r'\overline{\Q}', ring=data['end_ring_geom'] if is_curve else None))
data['real_geom_end_alg_name'] = real_geom_end_alg_name(curve['real_geom_end_alg'])
data['geom_end_alg_name'] = geom_end_alg_name(curve['geom_end_alg'])
data['end_alg_name'] = end_alg_name(curve['end_alg'])
# Endomorphism data over intermediate fields not already treated (only for curves, not necessarily isogeny invariant):
if is_curve:
data['end_lattice'] = (endo['lattice'])[1:-1]
if data['end_lattice']:
data['end_lattice_statement'] = end_lattice_statement(data['end_lattice'])
# Field over which the Jacobian decomposes (base field if Jacobian is geometrically simple)
data['is_simple_geom'] = endo['is_simple_geom']
data['split_field_label'] = endo['spl_fod_label']
data['split_field_poly'] = intlist_to_poly(endo['spl_fod_coeffs'])
data['split_field_statement'] = split_field_statement(data['is_simple_geom'], data['split_field_label'], data['split_field_poly'])
# Elliptic curve factors for non-simple Jacobians
if not data['is_simple_geom']:
data['split_coeffs'] = endo['spl_facs_coeffs']
if 'spl_facs_labels' in endo and len(endo['spl_facs_labels']) == len(endo['spl_facs_coeffs']):
data['split_labels'] = endo['spl_facs_labels']
data['split_condnorms'] = endo['spl_facs_condnorms']
data['split_statement'] = split_statement(data['split_coeffs'], data.get('split_labels'), data['split_condnorms'])
# Properties
self.properties = properties = [('Label', data['label'])]
if is_curve:
plot_from_db = db.g2c_plots.lucky({"label": curve['label']})
if (plot_from_db is None):
self.plot = encode_plot(eqn_list_to_curve_plot(data['min_eqn'], ratpts['rat_pts'] if ratpts else []))
else:
self.plot = plot_from_db['plot']
plot_link = '<a href="{0}"><img src="{0}" width="200" height="150"/></a>'.format(self.plot)
properties += [
(None, plot_link),
('Conductor', prop_int_pretty(data['cond'])),
('Discriminant', prop_int_pretty(data['disc'])),
]
if data['mw_rank_proved']:
properties += [('Mordell-Weil group', data['mw_group'])]
else:
properties += [('Conductor', prop_int_pretty(data['cond']))]
properties += [
('Sato-Tate group', data['st_group_link']),
(r'\(\End(J_{\overline{\Q}}) \otimes \R\)', r'\(%s\)' % data['real_geom_end_alg_name']),
(r'\(\End(J_{\overline{\Q}}) \otimes \Q\)', r'\(%s\)' % data['geom_end_alg_name']),
(r'\(\End(J) \otimes \Q\)', r'\(%s\)' % data['end_alg_name']),
(r'\(\overline{\Q}\)-simple', bool_pretty(data['is_simple_geom'])),
(r'\(\mathrm{GL}_2\)-type', bool_pretty(data['is_gl2_type'])),
]
# Friends
self.friends = friends = []
if is_curve:
friends.append(('Genus 2 curve %s.%s' % (data['slabel'][0], data['slabel'][1]), url_for(".by_url_isogeny_class_label", cond=data['slabel'][0], alpha=data['slabel'][1])))
# first deal with ECs and MFs
ecs = []
mfs = []
if 'split_labels' in data:
for friend_label in data['split_labels']:
if is_curve:
ecs.append(("Elliptic curve " + friend_label, url_for_ec(friend_label)))
else:
ecs.append(("Elliptic curve " + ec_label_class(friend_label), url_for_ec_class(friend_label)))
try:
cond, iso = ec_label_class(friend_label).split(".")
newform_label = ".".join([cond, str(2), 'a', iso])
mfs.append(("Modular form " + newform_label, url_for("cmf.by_url_newform_label", level=cond, weight=2, char_orbit_label='a', hecke_orbit=iso)))
except ValueError:
# means the friend isn't an elliptic curve over Q; adding Hilbert/Bianchi modular forms
# is dealt with via the L-functions instances below
pass
ecs.sort(key=lambda x: key_for_numerically_sort(x[0]))
mfs.sort(key=lambda x: key_for_numerically_sort(x[0]))
# then again EC from lfun
instances = []
for elt in db.lfunc_instances.search({'Lhash':data['Lhash'], 'type' : 'ECQP'}, 'url'):
instances.extend(elt.split('|'))
# and then the other isogeny friends
instances.extend([
elt['url'] for elt in
get_instances_by_Lhash_and_trace_hash(data["Lhash"],
4,
int(data["Lhash"])
)
])
exclude = {elt[1].rstrip('/').lstrip('/') for elt in self.friends
if elt[1]}
exclude.add(data['lfunc_url'].lstrip('/L/').rstrip('/'))
for elt in ecs + mfs + names_and_urls(instances, exclude=exclude):
# because of the splitting we must use G2C specific code
add_friend(friends, elt)
if is_curve:
friends.append(('Twists', url_for(".index_Q",
g20=str(data['g2'][0]),
g21=str(data['g2'][1]),
g22=str(data['g2'][2]))))
friends.append(('L-function', data['lfunc_url']))
# Breadcrumbs
self.bread = bread = [
('Genus 2 curves', url_for(".index")),
(r'$\Q$', url_for(".index_Q")),
('%s' % data['slabel'][0], url_for(".by_conductor", cond=data['slabel'][0])),
('%s' % data['slabel'][1], url_for(".by_url_isogeny_class_label", cond=data['slabel'][0], alpha=data['slabel'][1]))
]
if is_curve:
bread += [
('%s' % data['slabel'][2], url_for(".by_url_isogeny_class_discriminant", cond=data['slabel'][0], alpha=data['slabel'][1], disc=data['slabel'][2])),
('%s' % data['slabel'][3], url_for(".by_url_curve_label", cond=data['slabel'][0], alpha=data['slabel'][1], disc=data['slabel'][2], num=data['slabel'][3]))
]
# Title
self.title = "Genus 2 " + ("curve " if is_curve else "isogeny class ") + data['label']
# Code snippets (only for curves)
if not is_curve:
return
self.code = code = {}
code['show'] = {'sage':'','magma':''} # use default show names
f,h = fh = data['min_eqn']
g = simplify_hyperelliptic(fh)
code['curve'] = {'sage':'R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R(%s), R(%s));'%(f,h),
'magma':'R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R!%s, R!%s);'%(f,h) }
code['simple_curve'] = {'sage':'X = HyperellipticCurve(R(%s))'%(g), 'magma':'X,pi:= SimplifiedModel(C);' }
if data['abs_disc'] % 4096 == 0:
ind2 = [a[0] for a in data['bad_lfactors']].index(2)
bad2 = data['bad_lfactors'][ind2][1]
magma_cond_option = ': ExcFactors:=[*<2,Valuation('+str(data['cond'])+',2),R!'+str(bad2)+'>*]'
else:
magma_cond_option = ''
code['cond'] = {'magma': 'Conductor(LSeries(C%s)); Factorization($1);'% magma_cond_option}
code['disc'] = {'magma':'Discriminant(C); Factorization(Integers()!$1);'}
code['geom_inv'] = {'sage':'C.igusa_clebsch_invariants(); [factor(a) for a in _]',
'magma':'IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);'}
code['aut'] = {'magma':'AutomorphismGroup(C); IdentifyGroup($1);'}
code['autQbar'] = {'magma':'AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);'}
code['num_rat_wpts'] = {'magma':'#Roots(HyperellipticPolynomials(SimplifiedModel(C)));'}
if ratpts:
code['rat_pts'] = {'magma': '[' + ','.join(["C![%s,%s,%s]"%(p[0],p[1],p[2]) for p in ratpts['rat_pts']]) + ']; // minimal model'}
code['rat_pts_simp'] = {'magma': '[' + ','.join(["C![%s,%s,%s]"%(p[0],p[1],p[2]) for p in [simplify_hyperelliptic_point(data['min_eqn'], pt) for pt in ratpts['rat_pts']]]) + ']; // simplified model'}
code['mw_group'] = {'magma':'MordellWeilGroupGenus2(Jacobian(C));'}
code['two_selmer'] = {'magma':'TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);'}
code['has_square_sha'] = {'magma':'HasSquareSha(Jacobian(C));'}
code['locally_solvable'] = {'magma':'f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);'}
code['torsion_subgroup'] = {'magma':'TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);'}
def properties(self):
return [("Modulus", [prop_int_pretty(self.modulus)]),
("Structure", [self.structure]),
("Order", [prop_int_pretty(self.order)]),
]
def render_artin_representation_webpage(label):
if re.compile(r'^\d+$').match(label):
return artin_representation_search(**{'dimension': label, 'search_array': ArtinSearchArray()})
# label=dim.cond.nTt.indexcj, c is literal, j is index in conj class
# Should we have a big try around this to catch bad labels?
clean_label = clean_input(label)
if clean_label != label:
return redirect(url_for('.render_artin_representation_webpage', label=clean_label), 301)
# We could have a single representation or a Galois orbit
case = parse_any(label)
if case[0] == 'malformed':
try:
raise ValueError
except Exception:
flash_error("%s is not in a valid form for the label for an Artin representation or a Galois orbit of Artin representations", label)
return redirect(url_for(".index"))
# Do this twice to customize error messages
newlabel = case[1]
case = case[0]
if case == 'rep':
try:
the_rep = ArtinRepresentation(newlabel)
except Exception:
newlabel = parse_artin_label(label)
flash_error("Artin representation %s is not in database", label)
return redirect(url_for(".index"))
else: # it is an orbit
try:
the_rep = ArtinRepresentation(newlabel+'.a')
except Exception:
newlabel = parse_artin_orbit_label(newlabel)
flash_error("Galois orbit of Artin representations %s is not in database", label)
return redirect(url_for(".index"))
# in this case we want all characters
num_conj = the_rep.galois_conjugacy_size()
allchars = [ ArtinRepresentation(newlabel+'.'+num2letters(j)).character_formatted() for j in range(1,num_conj+1)]
label = newlabel
bread = get_bread([(artin_label_pretty(label), ' ')])
#artin_logger.info("Found %s" % (the_rep._data))
if case=='rep':
title = "Artin representation %s" % label
else:
title = "Galois orbit of Artin representations %s" % label
the_nf = the_rep.number_field_galois_group()
if the_rep.sign() == 0:
processed_root_number = "not computed"
else:
processed_root_number = '$%s$' % the_rep.sign()
properties = [("Label", artin_label_pretty(label)),
("Dimension", prop_int_pretty(the_rep.dimension())),
("Group", the_rep.group()),
("Conductor", prop_int_pretty(the_rep.conductor()))]
if case == 'rep':
properties.append( ("Root number", processed_root_number) )
properties.append( ("Indicator", prop_int_pretty(the_rep.indicator())) )
friends = []
wnf = None
proj_wnf = None
nf_url = the_nf.url_for()
if nf_url:
friends.append(("Field {}".format(the_nf.label()), nf_url))
wnf = the_nf.wnf()
proj_wnf = WebNumberField.from_coeffs(the_rep._data['Proj_Polynomial'])
if proj_wnf.is_in_db():
proj_coefs = [int(z) for z in proj_wnf.coeffs()]
if proj_coefs != the_nf.polynomial():
friends.append(("Field {}".format(proj_wnf.get_label()),
str(url_for("number_fields.by_label", label=proj_wnf.get_label()))))
if case == 'rep':
cc = the_rep.central_character()
if cc is not None:
if the_rep.dimension()==1:
if cc.order == 2:
cc_name = cc.symbol
else:
cc_name = cc.texname
friends.append(("Dirichlet character "+cc_name, url_for("characters.render_Dirichletwebpage", modulus=cc.modulus, number=cc.number)))
else:
detrep = the_rep.central_character_as_artin_rep()
friends.append(("Determinant "+detrep.label(), detrep.url_for()))
add_lfunction_friends(friends,label)
# once the L-functions are in the database, the link can always be shown
#if the_rep.dimension() <= 6:
if the_rep.dimension() == 1:
# Zeta is loaded differently
if cc.modulus == 1 and cc.number == 1:
friends.append(("L-function", url_for("l_functions.l_function_dirichlet_page", modulus=cc.modulus, number=cc.number)))
else:
# looking for Lhash dirichlet_L_modulus.number
mylhash = 'dirichlet_L_%d.%d'%(cc.modulus,cc.number)
lres = db.lfunc_instances.lucky({'Lhash': mylhash})
if lres is not None:
friends.append(("L-function", url_for("l_functions.l_function_dirichlet_page", modulus=cc.modulus, number=cc.number)))
# Dimension > 1
elif int(the_rep.conductor())**the_rep.dimension() <= 729000000000000:
friends.append(("L-function", url_for("l_functions.l_function_artin_page",
label=the_rep.label())))
orblabel = re.sub(r'\.[a-z]+$', '', label)
friends.append(("Galois orbit " + artin_label_pretty(orblabel),
url_for(".render_artin_representation_webpage", label=orblabel)))
else:
add_lfunction_friends(friends,label)
friends.append(("L-function", url_for("l_functions.l_function_artin_page", label=the_rep.label())))
for j in range(1,1+the_rep.galois_conjugacy_size()):
newlabel = label+'.'+num2letters(j)
friends.append(("Artin representation " + artin_label_pretty(newlabel),
url_for(".render_artin_representation_webpage", label=newlabel)))
info={} # for testing
if case == 'rep':
return render_template(
"artin-representation-show.html",
title=title,
bread=bread,
friends=friends,
object=the_rep,
cycle_string=cycle_string,
wnf=wnf,
proj_wnf=proj_wnf,
properties=properties,
info=info,
learnmore=learnmore_list(),
KNOWL_ID="artin.%s" % label,
)
# else we have an orbit
return render_template(
"artin-representation-galois-orbit.html",
title=title,
bread=bread,
allchars=allchars,
friends=friends,
object=the_rep,
cycle_string=cycle_string,
wnf=wnf,
proj_wnf=proj_wnf,
properties=properties,
info=info,
learnmore=learnmore_list(),
KNOWL_ID="artin.%s" % label,
)
def render_field_webpage(args):
data = None
info = {}
bread = bread_prefix()
# This function should not be called unless label is set.
label = clean_input(args['label'])
nf = WebNumberField(label)
data = {}
if nf.is_null():
if re.match(r'^\d+\.\d+\.\d+\.\d+$', label):
flash_error("Number field %s was not found in the database.",
label)
else:
flash_error("%s is not a valid label for a number field.", label)
return redirect(url_for(".number_field_render_webpage"))
info['wnf'] = nf
data['degree'] = nf.degree()
data['class_number'] = nf.class_number_latex()
ram_primes = nf.ramified_primes()
t = nf.galois_t()
n = nf.degree()
data['is_galois'] = nf.is_galois()
data['autstring'] = r'\Gal' if data['is_galois'] else r'\Aut'
data['is_abelian'] = nf.is_abelian()
if nf.is_abelian():
conductor = nf.conductor()
data['conductor'] = conductor
dirichlet_chars = nf.dirichlet_group()
if dirichlet_chars:
data['dirichlet_group'] = [
r'<a href = "%s">$\chi_{%s}(%s,·)$</a>' %
(url_for('characters.render_Dirichletwebpage',
modulus=data['conductor'],
number=j), data['conductor'], j)
for j in dirichlet_chars
]
if len(data['dirichlet_group']) == 1:
data[
'dirichlet_group'] = r'<span style="white-space:nowrap">$\lbrace$' + data[
'dirichlet_group'][0] + r'$\rbrace$</span>'
else:
data['dirichlet_group'] = r'$\lbrace$' + ', '.join(
data['dirichlet_group']
[:-1]) + '<span style="white-space:nowrap">' + data[
'dirichlet_group'][-1] + r'$\rbrace$</span>'
if data['conductor'].is_prime() or data['conductor'] == 1:
data['conductor'] = r"\(%s\)" % str(data['conductor'])
else:
factored_conductor = factor_base_factor(data['conductor'],
ram_primes)
factored_conductor = factor_base_factorization_latex(
factored_conductor)
data['conductor'] = r"\(%s=%s\)" % (str(
data['conductor']), factored_conductor)
data['galois_group'] = group_pretty_and_nTj(n, t, True)
data['auts'] = db.gps_transitive.lookup(r'{}T{}'.format(n, t))['auts']
data['cclasses'] = cclasses_display_knowl(n, t)
data['character_table'] = character_table_display_knowl(n, t)
data['class_group'] = nf.class_group()
data['class_group_invs'] = nf.class_group_invariants()
data['signature'] = nf.signature()
data['coefficients'] = nf.coeffs()
nf.make_code_snippets()
D = nf.disc()
data['disc_factor'] = nf.disc_factored_latex()
if D.abs().is_prime() or D == 1:
data['discriminant'] = bigint_knowl(D, cutoff=60, sides=3)
else:
data['discriminant'] = bigint_knowl(
D, cutoff=60,
sides=3) + r"\(\medspace = %s\)" % data['disc_factor']
if nf.frobs():
data['frob_data'], data['seeram'] = see_frobs(nf.frobs())
else: # fallback in case we haven't computed them in a case
data['frob_data'], data['seeram'] = frobs(nf)
# This could put commas in the rd, we don't want to trigger spaces
data['rd'] = ('$%s$' % fixed_prec(nf.rd(), 2)).replace(',', '{,}')
# Bad prime information
npr = len(ram_primes)
ramified_algebras_data = nf.ramified_algebras_data()
if isinstance(ramified_algebras_data, str):
loc_alg = ''
else:
# [label, latex, e, f, c, gal]
loc_alg = ''
for j in range(npr):
if ramified_algebras_data[j] is None:
loc_alg += '<tr><td>%s<td colspan="7">Data not computed' % str(
ram_primes[j]).rstrip('L')
else:
from lmfdb.local_fields.main import show_slope_content
mydat = ramified_algebras_data[j]
p = ram_primes[j]
loc_alg += '<tr><td rowspan="%d">$%s$</td>' % (len(mydat),
str(p))
mm = mydat[0]
myurl = url_for('local_fields.by_label', label=mm[0])
lab = mm[0]
if mm[3] * mm[2] == 1:
lab = r'$\Q_{%s}$' % str(p)
loc_alg += '<td><a href="%s">%s</a><td>$%s$<td>$%d$<td>$%d$<td>$%d$<td>%s<td>$%s$' % (
myurl, lab, mm[1], mm[2], mm[3], mm[4], mm[5],
show_slope_content(mm[8], mm[6], mm[7]))
for mm in mydat[1:]:
lab = mm[0]
myurl = url_for('local_fields.by_label', label=lab)
if mm[3] * mm[2] == 1:
lab = r'$\Q_{%s}$' % str(p)
loc_alg += '<tr><td><a href="%s">%s</a><td>$%s$<td>$%d$<td>$%d$<td>$%d$<td>%s<td>$%s$' % (
myurl, lab, mm[1], mm[2], mm[3], mm[4], mm[5],
show_slope_content(mm[8], mm[6], mm[7]))
loc_alg += '</tbody></table>'
ram_primes = str(ram_primes)[1:-1]
# Get rid of python L for big numbers
ram_primes = ram_primes.replace('L', '')
if not ram_primes:
ram_primes = r'\textrm{None}'
data['phrase'] = group_phrase(n, t)
zk = nf.zk()
Ra = PolynomialRing(QQ, 'a')
zk = [latex(Ra(x)) for x in zk]
zk = ['$%s$' % x for x in zk]
zk = ', '.join(zk)
grh_label = '<small>(<a title="assuming GRH" knowl="nf.assuming_grh">assuming GRH</a>)</small>' if nf.used_grh(
) else ''
# Short version for properties
grh_lab = nf.short_grh_string()
if 'computed' in str(data['class_number']):
grh_lab = ''
grh_label = ''
pretty_label = field_pretty(label)
if label != pretty_label:
pretty_label = "%s: %s" % (label, pretty_label)
info.update(data)
rootofunity = '%s (order $%d$)' % (nf.root_of_1_gen(),
nf.root_of_1_order())
info.update({
'label': pretty_label,
'label_raw': label,
'polynomial': web_latex(nf.poly()),
'ram_primes': ram_primes,
'integral_basis': zk,
'regulator': web_latex(nf.regulator()),
'unit_rank': nf.unit_rank(),
'root_of_unity': rootofunity,
'fund_units': nf.units_safe(),
'cnf': nf.cnf(),
'grh_label': grh_label,
'loc_alg': loc_alg
})
bread.append(('%s' % nf_label_pretty(info['label_raw']), ' '))
info['downloads_visible'] = True
info['downloads'] = [('worksheet', '/')]
info['friends'] = []
if nf.can_class_number():
# hide ones that take a lond time to compute on the fly
# note that the first degree 4 number field missed the zero of the zeta function
if abs(D**n) < 50000000:
info['friends'].append(('L-function', "/L/NumberField/%s" % label))
info['friends'].append(('Galois group', "/GaloisGroup/%dT%d" % (n, t)))
if 'dirichlet_group' in info:
info['friends'].append(('Dirichlet character group',
url_for("characters.dirichlet_group_table",
modulus=int(conductor),
char_number_list=','.join(
str(a) for a in dirichlet_chars),
poly=info['polynomial'])))
resinfo = []
galois_closure = nf.galois_closure()
if galois_closure[0] > 0:
if galois_closure[1]:
resinfo.append(('gc', galois_closure[1]))
if galois_closure[2]:
info['friends'].append(('Galois closure',
url_for(".by_label",
label=galois_closure[2][0])))
else:
resinfo.append(('gc', [dnc]))
sextic_twins = nf.sextic_twin()
if sextic_twins[0] > 0:
if sextic_twins[1]:
resinfo.append(('sex', r' $\times$ '.join(sextic_twins[1])))
else:
resinfo.append(('sex', dnc))
siblings = nf.siblings()
# [degsib list, label list]
# first is list of [deg, num expected, list of knowls]
if siblings[0]:
for sibdeg in siblings[0]:
if not sibdeg[2]:
sibdeg[2] = dnc
else:
nsibs = len(sibdeg[2])
sibdeg[2] = ', '.join(sibdeg[2])
if nsibs < sibdeg[1]:
sibdeg[2] += ', some ' + dnc
resinfo.append(('sib', siblings[0]))
for lab in siblings[1]:
if lab:
labparts = lab.split('.')
info['friends'].append(("Degree %s sibling" % labparts[0],
url_for(".by_label", label=lab)))
arith_equiv = nf.arith_equiv()
if arith_equiv[0] > 0:
if arith_equiv[1]:
resinfo.append(
('ae', ', '.join(arith_equiv[1]), len(arith_equiv[1])))
for aelab in arith_equiv[2]:
info['friends'].append(('Arithmetically equivalent sibling',
url_for(".by_label", label=aelab)))
else:
resinfo.append(('ae', dnc, len(arith_equiv[1])))
info['resinfo'] = resinfo
learnmore = learnmore_list()
title = "Number field %s" % info['label']
if npr == 1:
primes = 'prime'
else:
primes = 'primes'
if len(ram_primes) > 30:
ram_primes = 'see page'
else:
ram_primes = '$%s$' % ram_primes
properties = [('Label', nf_label_pretty(label)),
('Degree', prop_int_pretty(data['degree'])),
('Signature', '$%s$' % data['signature']),
('Discriminant', prop_int_pretty(D)),
('Root discriminant', '%s' % data['rd']),
('Ramified ' + primes + '', ram_primes),
('Class number', '%s %s' % (data['class_number'], grh_lab)),
('Class group',
'%s %s' % (data['class_group_invs'], grh_lab)),
('Galois group', group_pretty_and_nTj(data['degree'], t))]
downloads = [('Stored data to gp',
url_for('.nf_download', nf=label, download_type='data'))]
for lang in [["Magma", "magma"], ["SageMath", "sage"], ["Pari/GP", "gp"]]:
downloads.append(('Download {} code'.format(lang[0]),
url_for(".nf_download",
nf=label,
download_type=lang[1])))
from lmfdb.artin_representations.math_classes import NumberFieldGaloisGroup
from lmfdb.artin_representations.math_classes import artin_label_pretty
try:
info["tim_number_field"] = NumberFieldGaloisGroup(nf._data['coeffs'])
arts = [
z.label()
for z in info["tim_number_field"].artin_representations()
]
#print arts
for ar in arts:
info['friends'].append((
'Artin representation ' + artin_label_pretty(ar),
url_for(
"artin_representations.render_artin_representation_webpage",
label=ar)))
v = nf.factor_perm_repn(info["tim_number_field"])
def dopow(m):
if m == 0:
return ''
if m == 1:
return '*'
return '*<sup>%d</sup>' % m
info["mydecomp"] = [dopow(x) for x in v]
except AttributeError:
pass
return render_template("nf-show-field.html",
properties=properties,
credit=NF_credit,
title=title,
bread=bread,
code=nf.code,
friends=info.pop('friends'),
downloads=downloads,
learnmore=learnmore,
info=info,
KNOWL_ID="nf.%s" % label)
def make_class(self):
# Extract the size of the isogeny class from the database
classdata = db.ec_classdata.lucky({'lmfdb_iso': self.lmfdb_iso})
self.class_size = ncurves = classdata['class_size']
# Create a list of the curves in the class from the database
number_key = 'Cnumber' if self.label_type == 'Cremona' else 'lmfdb_number'
self.curves = [
db.ec_curvedata.lucky({
'lmfdb_iso': self.lmfdb_iso,
number_key: i + 1
}) for i in range(ncurves)
]
# Set optimality flags. The optimal curve is conditionally
# number 1 except in one case which is labeled differently in
# the Cremona tables. We know which curve is optimal iff the
# optimality code for curve #1 is 1 (except for class 990h).
# Note that self is actually an elliptic curve, with number=1.
# The code here allows us to update the display correctly by
# changing one line in this file (defining OPTIMALITY_BOUND)
# without changing the data.
self.cremona_bound = CREMONA_BOUND
self.optimality_bound = OPTIMALITY_BOUND
self.optimality_known = (self.conductor < OPTIMALITY_BOUND) or (
(self.conductor < CREMONA_BOUND) and ((self.optimality == 1) or
(self.Ciso == '990h')))
self.optimal_label = self.Clabel if self.label_type == 'Cremona' else self.lmfdb_label
if self.conductor < OPTIMALITY_BOUND:
for c in self.curves:
c['optimal'] = (c['Cnumber'] == (3 if self.Ciso == '990h' else
1))
c['optimality_known'] = True
elif self.conductor < CREMONA_BOUND:
for c in self.curves:
c['optimal'] = (c['optimality'] > 0
) # this curve possibly optimal
c['optimality_known'] = (c['optimality'] == 1
) # this curve certainly optimal
else:
for c in self.curves:
c['optimal'] = None
c['optimality_known'] = False
for c in self.curves:
c['ai'] = c['ainvs']
c['curve_url_lmfdb'] = url_for(".by_triple_label",
conductor=self.conductor,
iso_label=self.iso_label,
number=c['lmfdb_number'])
c['curve_url_cremona'] = url_for(
".by_ec_label",
label=c['Clabel']) if self.conductor < CREMONA_BOUND else "N/A"
if self.label_type == 'Cremona':
c['curve_label'] = c['Clabel']
_, c_iso, c_number = split_cremona_label(c['Clabel'])
else:
c['curve_label'] = c['lmfdb_label']
_, c_iso, c_number = split_lmfdb_label(c['lmfdb_label'])
c['short_label'] = "{}{}".format(c_iso, c_number)
from sage.matrix.all import Matrix
M = classdata['isogeny_matrix']
# permute rows/cols to match labelling: the rows/cols in the
# ec_classdata table are with respect to LMFDB ordering.
if self.label_type == 'Cremona':
perm = lambda i: next(c for c in self.curves
if c['Cnumber'] == i + 1)['lmfdb_number'] - 1
M = [[M[perm(i)][perm(j)] for i in range(ncurves)]
for j in range(ncurves)]
M = Matrix(M)
self.isogeny_matrix_str = latex(M)
# Create isogeny graph with appropriate vertex labels:
self.graph = make_graph(M, [c['short_label'] for c in self.curves])
P = self.graph.plot(edge_labels=True, vertex_size=1000)
self.graph_img = encode_plot(P)
self.graph_link = '<img src="%s" width="200" height="150"/>' % self.graph_img
self.newform = raw_typeset(
PowerSeriesRing(QQ, 'q')(classdata['anlist'], 20, check=True))
self.newform_label = ".".join(
[str(self.conductor),
str(2), 'a', self.iso_label])
self.newform_exists_in_db = db.mf_newforms.label_exists(
self.newform_label)
if self.newform_exists_in_db:
char_orbit, hecke_orbit = self.newform_label.split('.')[2:]
self.newform_link = url_for("cmf.by_url_newform_label",
level=self.conductor,
weight=2,
char_orbit_label=char_orbit,
hecke_orbit=hecke_orbit)
self.lfunction_link = url_for("l_functions.l_function_ec_page",
conductor_label=self.conductor,
isogeny_class_label=self.iso_label)
self.friends = [('L-function', self.lfunction_link)]
if self.cm:
# set CM field for Properties box.
D = ZZ(self.cm).squarefree_part()
coeffs = [(1 - D) // 4, -1, 1] if D % 4 == 1 else [-D, 0, 1]
lab = db.nf_fields.lucky({'coeffs': coeffs}, projection='label')
self.CMfield = field_pretty(lab)
else:
self.CMfield = "no"
if self.conductor <= 300:
self.friends += [('Symmetric square L-function',
url_for("l_functions.l_function_ec_sym_page",
power='2',
conductor=self.conductor,
isogeny=self.iso_label))]
if self.conductor <= 50:
self.friends += [('Symmetric cube L-function',
url_for("l_functions.l_function_ec_sym_page",
power='3',
conductor=self.conductor,
isogeny=self.iso_label))]
if self.newform_exists_in_db:
self.friends += [('Modular form ' + self.newform_label,
self.newform_link)]
if self.label_type == 'Cremona':
self.title = "Elliptic curve isogeny class with Cremona label {} (LMFDB label {})".format(
self.Ciso, self.lmfdb_iso)
elif self.conductor < CREMONA_BOUND:
self.title = "Elliptic curve isogeny class with LMFDB label {} (Cremona label {})".format(
self.lmfdb_iso, self.Ciso)
else:
self.title = "Elliptic curve isogeny class with LMFDB label {}".format(
self.lmfdb_iso)
self.properties = [
('Label',
self.Ciso if self.label_type == 'Cremona' else self.lmfdb_iso),
('Number of curves', prop_int_pretty(ncurves)),
('Conductor', prop_int_pretty(self.conductor)),
('CM', '%s' % self.CMfield), ('Rank', prop_int_pretty(self.rank))
]
if ncurves > 1:
self.properties += [('Graph', ''), (None, self.graph_link)]
self.downloads = [('q-expansion to text',
url_for(".download_EC_qexp",
label=self.iso_label,
limit=1000)),
('All stored data to text',
url_for(".download_EC_all", label=self.iso_label))]
self.bread = [('Elliptic curves', url_for("ecnf.index")),
(r'$\Q$', url_for(".rational_elliptic_curves")),
('%s' % self.conductor,
url_for(".by_conductor", conductor=self.conductor)),
('%s' % self.iso_label, ' ')]
self.code = {}
self.code['show'] = {'sage': ''} # use default show names
self.code['class'] = {
'sage':
'E = EllipticCurve("%s1")\n' % (self.iso_label) +
'E.isogeny_class()\n'
}
self.code['curves'] = {'sage': 'E.isogeny_class().curves'}
self.code['rank'] = {'sage': 'E.rank()'}
self.code['q_eigenform'] = {'sage': 'E.q_eigenform(10)'}
self.code['matrix'] = {'sage': 'E.isogeny_class().matrix()'}
self.code['plot'] = {
'sage': 'E.isogeny_graph().plot(edge_labels=True)'
}
def make_curve(self):
data = self.data = {}
lmfdb_label = self.lmfdb_label
# Some data fields of self are just those from the database.
# These only need some reformatting.
data['ainvs'] = self.ainvs
data['conductor'] = N = self.conductor
data['j_invariant'] = QQ(tuple(self.jinv))
data['j_inv_factor'] = latex(0)
if data['j_invariant']: # don't factor 0
data['j_inv_factor'] = latex(data['j_invariant'].factor())
data['j_inv_latex'] = web_latex(data['j_invariant'])
# retrieve local reduction data from table ec_localdata:
self.local_data = local_data = list(
db.ec_localdata.search({"lmfdb_label": lmfdb_label}))
for ld in local_data:
if ld['kodaira_symbol'] <= -14:
# Work around bug in Sage's latex
ld['kod'] = 'I_{%s}^{*}' % (-ld['kodaira_symbol'] - 4)
else:
ld['kod'] = latex(KodairaSymbol(ld['kodaira_symbol']))
Nfac = Factorization([(ZZ(ld['prime']), ld['conductor_valuation'])
for ld in local_data])
Dfac = Factorization([(ZZ(ld['prime']), ld['discriminant_valuation'])
for ld in local_data],
unit=ZZ(self.signD))
data['disc_factor'] = latex(Dfac)
data['disc'] = D = Dfac.value()
data['cond_factor'] = latex(Nfac)
data['disc_latex'] = web_latex(D)
data['cond_latex'] = web_latex(N)
# retrieve data about MW rank, generators, heights and
# torsion, leading term of L-function & other BSD data from
# table ec_mwbsd:
self.make_mwbsd()
# latex equation:
latexeqn = latex_equation(self.ainvs)
data['equation'] = raw_typeset(unlatex(latexeqn), latexeqn)
# minimal quadratic twist:
data['minq_D'] = minqD = self.min_quad_twist_disc
data['minq_label'] = db.ec_curvedata.lucky(
{'ainvs': self.min_quad_twist_ainvs},
projection='lmfdb_label'
if self.label_type == 'LMFDB' else 'Clabel')
data['minq_info'] = '(itself)' if minqD == 1 else '(by {})'.format(
minqD)
# modular degree:
try:
data['degree'] = ZZ(self.degree) # convert None to 0
except AttributeError: # if not computed, db has Null and the attribute is missing
data['degree'] = 0 # invalid, but will be displayed nicely
# coefficients of modular form / L-series:
classdata = db.ec_classdata.lookup(self.lmfdb_iso)
data['an'] = classdata['anlist']
data['ap'] = classdata['aplist']
# mod-p Galois images:
data['galois_data'] = list(
db.ec_galrep.search({'lmfdb_label': lmfdb_label}))
for gd in data[
'galois_data']: # remove the prime prefix from each image code
gd['image'] = trim_galois_image_code(gd['image'])
# CM and Endo ring:
data['CMD'] = self.cm
data['CM'] = "no"
data['EndE'] = r"\(\Z\)"
if self.cm:
data['cm_ramp'] = [
p for p in ZZ(self.cm).support() if not p in self.nonmax_primes
]
data['cm_nramp'] = len(data['cm_ramp'])
if data['cm_nramp'] == 1:
data['cm_ramp'] = data['cm_ramp'][0]
else:
data['cm_ramp'] = ", ".join(str(p) for p in data['cm_ramp'])
data['cm_sqf'] = ZZ(self.cm).squarefree_part()
data['CM'] = r"yes (\(D=%s\))" % data['CMD']
if data['CMD'] % 4 == 0:
d4 = ZZ(data['CMD']) // 4
data['EndE'] = r"\(\Z[\sqrt{%s}]\)" % d4
else:
data['EndE'] = r"\(\Z[(1+\sqrt{%s})/2]\)" % data['CMD']
data['ST'] = st_link_by_name(1, 2, 'N(U(1))')
else:
data['ST'] = st_link_by_name(1, 2, 'SU(2)')
# Isogeny degrees:
cond, iso, num = split_lmfdb_label(lmfdb_label)
self.class_deg = classdata['class_deg']
self.one_deg = ZZ(self.class_deg).is_prime()
isodegs = [str(d) for d in self.isogeny_degrees if d > 1]
if len(isodegs) < 3:
data['isogeny_degrees'] = " and ".join(isodegs)
else:
data['isogeny_degrees'] = " and ".join(
[", ".join(isodegs[:-1]), isodegs[-1]])
self.make_twoadic_data()
# Optimality
# The optimal curve in the class is the curve whose Cremona
# label ends in '1' except for '990h' which was labelled
# wrongly long ago. This is proved for N up to
# OPTIMALITY_BOUND (and when there is only one curve in an
# isogeny class, obviously) and expected for all N.
# Column 'optimality' is 1 for certainly optimal curves, 0 for
# certainly non-optimal curves, and is n>1 if the curve is one
# of n in the isogeny class which may be optimal given current
# knowledge.
# Column "manin_constant' is the correct Manin constant
# assuming that the optimal curve in the class is known, or
# otherwise if it is the curve with (Cremona) number 1.
# The code here allows us to update the display correctly by
# changing one line in this file (defining OPTIMALITY_BOUND)
# without changing the data.
data['optimality_bound'] = OPTIMALITY_BOUND
self.cremona_bound = CREMONA_BOUND
if N < CREMONA_BOUND:
data[
'manin_constant'] = self.manin_constant # (conditional on data['optimality_known'])
else:
data['manin_constant'] = 0 # (meaning not available)
if N < OPTIMALITY_BOUND:
data['optimality_code'] = int(
self.Cnumber == (3 if self.Ciso == '990h' else 1))
data['optimality_known'] = True
data['manin_known'] = True
if self.label_type == 'Cremona':
data[
'optimal_label'] = '990h3' if self.Ciso == '990h' else self.Ciso + '1'
else:
data[
'optimal_label'] = '990.i3' if self.lmfdb_iso == '990.i' else self.lmfdb_iso + '1'
elif N < CREMONA_BOUND:
data['optimality_code'] = self.optimality
data['optimality_known'] = (self.optimality < 2)
if self.optimality == 1:
data['manin_known'] = True
data[
'optimal_label'] = self.Clabel if self.label_type == 'Cremona' else self.lmfdb_label
else:
if self.Cnumber == 1:
data['manin_known'] = False
data[
'optimal_label'] = self.Clabel if self.label_type == 'Cremona' else self.lmfdb_label
else:
# find curve #1 in this class and its optimailty code:
opt_curve = db.ec_curvedata.lucky(
{
'Ciso': self.Ciso,
'Cnumber': 1
},
projection=['Clabel', 'lmfdb_label', 'optimality'])
data['manin_known'] = (opt_curve['optimality'] == 1)
data['optimal_label'] = opt_curve[
'Clabel' if self.label_type ==
'Cremona' else 'lmfdb_label']
else:
data['optimality_code'] = None
data['optimality_known'] = False
data['manin_known'] = False
data['optimal_label'] = ''
# p-adic data:
data['p_adic_primes'] = [
p for i, p in enumerate(prime_range(5, 100))
if (N * data['ap'][i]) % p != 0
]
data['p_adic_data_exists'] = False
if data['optimality_code'] == 1:
data['p_adic_data_exists'] = db.ec_padic.exists(
{'lmfdb_iso': self.lmfdb_iso})
# Iwasawa data (where present)
self.make_iwasawa()
# Torsion growth data (where present)
self.make_torsion_growth()
# Newform
rawnewform = str(PowerSeriesRing(QQ, 'q')(data['an'], 20, check=True))
data['newform'] = raw_typeset(
rawnewform,
web_latex(PowerSeriesRing(QQ, 'q')(data['an'], 20, check=True)))
data['newform_label'] = self.newform_label = ".".join(
[str(cond), str(2), 'a', iso])
self.newform_link = url_for("cmf.by_url_newform_label",
level=cond,
weight=2,
char_orbit_label='a',
hecke_orbit=iso)
self.newform_exists_in_db = db.mf_newforms.label_exists(
self.newform_label)
self._code = None
if self.label_type == 'Cremona':
self.class_url = url_for(".by_ec_label", label=self.Ciso)
self.class_name = self.Ciso
else:
self.class_url = url_for(".by_ec_label", label=self.lmfdb_iso)
self.class_name = self.lmfdb_iso
data['class_name'] = self.class_name
data['Cnumber'] = self.Cnumber if N < CREMONA_BOUND else None
self.friends = [('Isogeny class ' + self.class_name, self.class_url),
('Minimal quadratic twist %s %s' %
(data['minq_info'], data['minq_label']),
url_for(".by_ec_label", label=data['minq_label'])),
('All twists ',
url_for(".rational_elliptic_curves",
jinv=data['j_invariant']))]
lfun_url = url_for("l_functions.l_function_ec_page",
conductor_label=N,
isogeny_class_label=iso)
origin_url = lfun_url.lstrip('/L/').rstrip('/')
if db.lfunc_instances.exists({'url': origin_url}):
self.friends += [('L-function', lfun_url)]
else:
self.friends += [('L-function not available', "")]
if not self.cm:
if N <= 300:
self.friends += [('Symmetric square L-function',
url_for("l_functions.l_function_ec_sym_page",
power='2',
conductor=N,
isogeny=iso))]
if N <= 50:
self.friends += [('Symmetric cube L-function',
url_for("l_functions.l_function_ec_sym_page",
power='3',
conductor=N,
isogeny=iso))]
if self.newform_exists_in_db:
self.friends += [('Modular form ' + self.newform_label,
self.newform_link)]
self.downloads = [('q-expansion to text',
url_for(".download_EC_qexp",
label=self.lmfdb_label,
limit=1000)),
('All stored data to text',
url_for(".download_EC_all",
label=self.lmfdb_label)),
('Code to Magma',
url_for(".ec_code_download",
conductor=cond,
iso=iso,
number=num,
label=self.lmfdb_label,
download_type='magma')),
('Code to SageMath',
url_for(".ec_code_download",
conductor=cond,
iso=iso,
number=num,
label=self.lmfdb_label,
download_type='sage')),
('Code to GP',
url_for(".ec_code_download",
conductor=cond,
iso=iso,
number=num,
label=self.lmfdb_label,
download_type='gp'))]
try:
self.plot = encode_plot(self.E.plot())
except AttributeError:
self.plot = encode_plot(EllipticCurve(data['ainvs']).plot())
self.plot_link = '<a href="{0}"><img src="{0}" width="200" height="150"/></a>'.format(
self.plot)
self.properties = [
('Label', self.Clabel
if self.label_type == 'Cremona' else self.lmfdb_label),
(None, self.plot_link),
('Conductor', prop_int_pretty(data['conductor'])),
('Discriminant', prop_int_pretty(data['disc'])),
('j-invariant', '%s' % data['j_inv_latex']),
('CM', '%s' % data['CM']),
('Rank', 'unknown' if self.mwbsd['rank'] == '?' else
prop_int_pretty(self.mwbsd['rank'])),
('Torsion structure', (r'\(%s\)' % self.mwbsd['tor_struct'])
if self.mwbsd['tor_struct'] else 'trivial'),
]
if self.label_type == 'Cremona':
self.title = "Elliptic curve with Cremona label {} (LMFDB label {})".format(
self.Clabel, self.lmfdb_label)
elif N < CREMONA_BOUND:
self.title = "Elliptic curve with LMFDB label {} (Cremona label {})".format(
self.lmfdb_label, self.Clabel)
else:
self.title = "Elliptic curve with LMFDB label {}".format(
self.lmfdb_label)
self.bread = [('Elliptic curves', url_for("ecnf.index")),
(r'$\Q$', url_for(".rational_elliptic_curves")),
('%s' % N, url_for(".by_conductor", conductor=N)),
('%s' % iso,
url_for(".by_double_iso_label",
conductor=N,
iso_label=iso)), ('%s' % num, ' ')]
def make_class(self):
self.CM = self.cm
N, iso, number = split_lmfdb_label(self.lmfdb_iso)
self.conductor = N = int(N)
# Extract the size of the isogeny class from the database
self.ncurves = ncurves = self.class_size
# Create a list of the curves in the class from the database
number_key = 'number' if self.label_type == 'Cremona' else 'lmfdb_number'
self.curves = [
db.ec_curves.lucky({
'iso': self.iso,
number_key: i + 1
}) for i in range(ncurves)
]
# Set optimality flags. The optimal curve is conditionally
# number 1 except in one case which is labeled differently in
# the Cremona tables. We know which curve is optimal iff the
# optimality code for curve #1 is 1 (except for class 990h).
# Note that self is actually an elliptic curve, with number=1.
# The code here allows us to update the display correctly by
# changing one line in this file (defining OPTIMALITY_BOUND)
# without changing the data.
self.optimality_bound = OPTIMALITY_BOUND
self.optimality_known = (N < OPTIMALITY_BOUND) or (self.optimality
== 1) or (self.iso
== '990h')
self.optimal_label = self.label if self.label_type == 'Cremona' else self.lmfdb_label
if N < OPTIMALITY_BOUND:
for c in self.curves:
c['optimal'] = (c['number'] == (3
if self.iso == '990h' else 1))
c['optimality_known'] = True
else:
for c in self.curves:
c['optimal'] = (c['optimality'] > 0
) # this curve possibly optimal
c['optimality_known'] = (c['optimality'] == 1
) # this curve certainly optimal
for c in self.curves:
c['ai'] = c['ainvs']
c['curve_url_lmfdb'] = url_for(".by_triple_label",
conductor=N,
iso_label=iso,
number=c['lmfdb_number'])
c['curve_url_cremona'] = url_for(".by_ec_label", label=c['label'])
if self.label_type == 'Cremona':
c['curve_label'] = c['label']
_, c_iso, c_number = split_cremona_label(c['label'])
else:
c['curve_label'] = c['lmfdb_label']
_, c_iso, c_number = split_lmfdb_label(c['lmfdb_label'])
c['short_label'] = "{}{}".format(c_iso, c_number)
from sage.matrix.all import Matrix
if self.label_type == 'Cremona':
# permute rows/cols
perm = lambda i: next(c for c in self.curves
if c['number'] == i + 1)['lmfdb_number'] - 1
self.isogeny_matrix = [[
self.isogeny_matrix[perm(i)][perm(j)] for i in range(ncurves)
] for j in range(ncurves)]
self.isogeny_matrix = Matrix(self.isogeny_matrix)
self.isogeny_matrix_str = latex(matrix(self.isogeny_matrix))
# Create isogeny graph with appropriate vertex labels:
self.graph = make_graph(self.isogeny_matrix,
[c['short_label'] for c in self.curves])
P = self.graph.plot(edge_labels=True, vertex_size=1000)
self.graph_img = encode_plot(P)
self.graph_link = '<img src="%s" width="200" height="150"/>' % self.graph_img
self.newform = web_latex(
PowerSeriesRing(QQ, 'q')(self.anlist, 20, check=True))
self.newform_label = ".".join([str(N), str(2), 'a', iso])
# self.newform_label = db.mf_newforms.lucky({'level':N, 'weight':2, 'related_objects':{'$contains':'EllipticCurve/Q/%s/%s' % (N, iso)}},'label')
# self.newform_exists_in_db = self.newform_label is not None
self.newform_exists_in_db = db.mf_newforms.label_exists(
self.newform_label)
# if self.newform_label is not None:
if self.newform_exists_in_db:
char_orbit, hecke_orbit = self.newform_label.split('.')[2:]
self.newform_link = url_for("cmf.by_url_newform_label",
level=N,
weight=2,
char_orbit_label=char_orbit,
hecke_orbit=hecke_orbit)
self.lfunction_link = url_for("l_functions.l_function_ec_page",
conductor_label=N,
isogeny_class_label=iso)
self.friends = [('L-function', self.lfunction_link)]
if not self.CM:
self.CM = "no"
if int(N) <= 300:
self.friends += [('Symmetric square L-function',
url_for("l_functions.l_function_ec_sym_page",
power='2',
conductor=N,
isogeny=iso))]
if int(N) <= 50:
self.friends += [('Symmetric cube L-function',
url_for("l_functions.l_function_ec_sym_page",
power='3',
conductor=N,
isogeny=iso))]
if self.newform_exists_in_db:
self.friends += [('Modular form ' + self.newform_label,
self.newform_link)]
if self.label_type == 'Cremona':
self.title = "Elliptic curve isogeny class with Cremona label {} (LMFDB label {})".format(
self.iso, self.lmfdb_iso)
self.iso_label = self.iso
else:
self.title = "Elliptic curve isogeny class with LMFDB label {} (Cremona label {})".format(
self.lmfdb_iso, self.iso)
self.iso_label = self.lmfdb_iso
self.properties = [
('Label',
self.iso if self.label_type == 'Cremona' else self.lmfdb_iso),
('Number of curves', prop_int_pretty(ncurves)),
('Conductor', prop_int_pretty(N)), ('CM', '%s' % self.CM),
('Rank', prop_int_pretty(self.rank))
]
if self.ncurves > 1:
self.properties += [('Graph', ''), (None, self.graph_link)]
self.downloads = [('q-expansion to text',
url_for(".download_EC_qexp",
label=self.lmfdb_iso,
limit=1000)),
('All stored data to text',
url_for(".download_EC_all", label=self.lmfdb_iso))]
self.bread = [('Elliptic curves', url_for("ecnf.index")),
(r'$\Q$', url_for(".rational_elliptic_curves")),
('%s' % N, url_for(".by_conductor", conductor=N)),
('%s' % iso, ' ')]
self.code = {}
self.code['show'] = {'sage': ''} # use default show names
self.code['class'] = {
'sage':
'E = EllipticCurve("%s1")\n' % (self.lmfdb_iso) +
'E.isogeny_class()\n'
}
self.code['curves'] = {'sage': 'E.isogeny_class().curves'}
self.code['rank'] = {'sage': 'E.rank()'}
self.code['q_eigenform'] = {'sage': 'E.q_eigenform(10)'}
self.code['matrix'] = {'sage': 'E.isogeny_class().matrix()'}
self.code['plot'] = {
'sage': 'E.isogeny_graph().plot(edge_labels=True)'
}