class WebEC(object):
"""
Class for an elliptic curve over Q
"""
def __init__(self, dbdata):
"""
Arguments:
- dbdata: the data from the database
"""
logger.debug("Constructing an instance of ECisog_class")
self.__dict__.update(dbdata)
# Next lines because the hyphens make trouble
self.xintcoords = parse_list(dbdata['x-coordinates_of_integral_points'])
self.non_surjective_primes = dbdata['non-surjective_primes']
# Next lines because the python identifiers cannot start with 2
self.twoadic_index = dbdata['2adic_index']
self.twoadic_log_level = dbdata['2adic_log_level']
self.twoadic_gens = dbdata['2adic_gens']
self.twoadic_label = dbdata['2adic_label']
# All other fields are handled here
self.make_curve()
@staticmethod
def by_label(label):
"""
Searches for a specific elliptic curve in the curves
collection by its label, which can be either in LMFDB or
Cremona format.
"""
try:
N, iso, number = split_lmfdb_label(label)
data = db_ec().find_one({"lmfdb_label" : label})
except AttributeError:
try:
N, iso, number = split_cremona_label(label)
data = db_ec().find_one({"label" : label})
except AttributeError:
return "Invalid label" # caller must catch this and raise an error
if data:
return WebEC(data)
return "Curve not found" # caller must catch this and raise an error
def make_curve(self):
# To start with the data fields of self are just those from
# the database. We need to reformat these, construct the
# actual elliptic curve E, and compute some further (easy)
# data about it.
#
# Weierstrass equation
data = self.data = {}
data['ainvs'] = [int(ai) for ai in self.ainvs]
self.E = EllipticCurve(data['ainvs'])
data['equation'] = web_latex(self.E)
# conductor, j-invariant and discriminant
data['conductor'] = N = ZZ(self.conductor)
bad_primes = N.prime_factors()
try:
data['j_invariant'] = QQ(str(self.jinv))
except KeyError:
data['j_invariant'] = self.E.j_invariant()
data['j_inv_factor'] = latex(0)
if data['j_invariant']:
data['j_inv_factor'] = latex(data['j_invariant'].factor())
data['j_inv_str'] = unicode(str(data['j_invariant']))
data['j_inv_latex'] = web_latex(data['j_invariant'])
data['disc'] = D = self.E.discriminant()
data['disc_latex'] = web_latex(data['disc'])
data['disc_factor'] = latex(data['disc'].factor())
data['cond_factor'] =latex(N.factor())
data['cond_latex'] = web_latex(N)
# CM and endomorphism ring
data['CMD'] = self.cm
data['CM'] = "no"
data['EndE'] = "\(\Z\)"
if self.cm:
data['CM'] = "yes (\(D=%s\))" % data['CMD']
if data['CMD']%4==0:
d4 = ZZ(data['CMD'])//4
data['EndE'] = "\(\Z[\sqrt{%s}]\)" % d4
else:
data['EndE'] = "\(\Z[(1+\sqrt{%s})/2]\)" % data['CMD']
# modular degree
try:
data['degree'] = self.degree
except AttributeError:
try:
data['degree'] = self.E.modular_degree()
except RuntimeError:
data['degree'] # invalid, but will be displayed nicely
# Minimal quadratic twist
E_pari = self.E.pari_curve()
from sage.libs.pari.all import PariError
try:
minq, minqD = self.E.minimal_quadratic_twist()
except PariError: # this does occur with 164411a1
ec.debug("PariError computing minimal quadratic twist of elliptic curve %s" % lmfdb_label)
minq = self.E
minqD = 1
data['minq_D'] = minqD
if self.E == minq:
data['minq_label'] = self.lmfdb_label
data['minq_info'] = '(itself)'
else:
minq_ainvs = [str(c) for c in minq.ainvs()]
data['minq_label'] = db_ec().find_one({'ainvs': minq_ainvs})['lmfdb_label']
data['minq_info'] = '(by %s)' % minqD
minq_N, minq_iso, minq_number = split_lmfdb_label(data['minq_label'])
# rational and integral points
mw = self.mw = {}
xintpoints_projective = [self.E.lift_x(x) for x in self.xintcoords]
xintpoints = [P.xy() for P in xintpoints_projective]
mw['int_points'] = ', '.join(web_latex(P) for P in xintpoints)
# Generators of infinite order
mw['rank'] = self.rank
try:
self.generators = [self.E(g) for g in parse_points(self.gens)]
mw['generators'] = [web_latex(P.xy()) for P in self.generators]
mw['heights'] = [P.height() for P in self.generators]
except AttributeError:
mw['generators'] = ''
mw['heights'] = []
# Torsion subgroup: order, structure, generators
mw['tor_order'] = self.torsion
tor_struct = [int(c) for c in self.torsion_structure]
if mw['tor_order'] == 1:
mw['tor_struct'] = '\mathrm{Trivial}'
mw['tor_gens'] = ''
else:
mw['tor_struct'] = ' \\times '.join(['\Z/{%s}\Z' % n
for n in tor_struct])
mw['tor_gens'] = ', '.join(web_latex(self.E(g).xy()) for g in parse_points(self.torsion_generators))
# Images of Galois representations
try:
data['galois_images'] = [trim_galois_image_code(s) for s in self.galois_images]
data['non_surjective_primes'] = self.non_surjective_primes
except AttributeError:
#print "No Galois image data"
data['galois_images'] = []
data['non_surjective_primes'] = []
data['galois_data'] = [{'p': p,'image': im }
for p,im in zip(data['non_surjective_primes'],
data['galois_images'])]
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 & BSD data
bsd = self.bsd = {}
r = self.rank
if r >= 2:
bsd['lder_name'] = "L^{(%s)}(E,1)/%s!" % (r,r)
elif r:
bsd['lder_name'] = "L'(E,1)"
else:
bsd['lder_name'] = "L(E,1)"
bsd['reg'] = self.regulator
bsd['omega'] = self.real_period
bsd['sha'] = int(0.1+self.sha_an)
bsd['lder'] = self.special_value
# 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)
if self.iso == '990h':
data['Gamma0optimal'] = bool(self.number == 3)
else:
data['Gamma0optimal'] = bool(self.number == 1)
data['p_adic_data_exists'] = False
if data['Gamma0optimal']:
data['p_adic_data_exists'] = (padic_db().find({'lmfdb_iso': self.lmfdb_iso}).count()) > 0
data['p_adic_primes'] = [p for p in sage.all.prime_range(5, 100)
if self.E.is_ordinary(p) and not p.divides(N)]
# Local data
local_data = self.local_data = []
# if we use E.tamagawa_numbers() it calls E.local_data(p) which
# used to crash on some curves e.g. 164411a1
tamagawa_numbers = []
for p in bad_primes:
local_info = self.E.local_data(p, algorithm="generic")
local_data_p = {}
local_data_p['p'] = p
local_data_p['tamagawa_number'] = local_info.tamagawa_number()
tamagawa_numbers.append(ZZ(local_info.tamagawa_number()))
local_data_p['kodaira_symbol'] = web_latex(local_info.kodaira_symbol()).replace('$', '')
local_data_p['reduction_type'] = local_info.bad_reduction_type()
local_data_p['ord_cond'] = local_info.conductor_valuation()
local_data_p['ord_disc'] = local_info.discriminant_valuation()
local_data_p['ord_den_j'] = max(0,-self.E.j_invariant().valuation(p))
local_data.append(local_data_p)
if len(bad_primes)>1:
bsd['tamagawa_factors'] = r' \cdot '.join(str(c.factor()) for c in tamagawa_numbers)
else:
bsd['tamagawa_factors'] = ''
bsd['tamagawa_product'] = sage.misc.all.prod(tamagawa_numbers)
cond, iso, num = split_lmfdb_label(self.lmfdb_label)
data['newform'] = web_latex(self.E.q_eigenform(10))
self.make_code_snippets()
self.friends = [
('Isogeny class ' + self.lmfdb_iso, url_for(".by_double_iso_label", conductor=N, iso_label=iso)),
('Minimal quadratic twist %s %s' % (data['minq_info'], data['minq_label']), url_for(".by_triple_label", conductor=minq_N, iso_label=minq_iso, number=minq_number)),
('All twists ', url_for(".rational_elliptic_curves", jinv=self.jinv)),
('L-function', url_for("l_functions.l_function_ec_page", label=self.lmfdb_label)),
('Symmetric square L-function', url_for("l_functions.l_function_ec_sym_page", power='2', label=self.lmfdb_iso)),
('Symmetric 4th power L-function', url_for("l_functions.l_function_ec_sym_page", power='4', label=self.lmfdb_iso)),
('Modular form ' + self.lmfdb_iso.replace('.', '.2'), url_for("emf.render_elliptic_modular_forms", level=int(N), weight=2, character=0, label=iso))]
self.downloads = [('Download coeffients of q-expansion', url_for(".download_EC_qexp", label=self.lmfdb_label, limit=100)),
('Download all stored data', url_for(".download_EC_all", label=self.lmfdb_label))]
self.plot = encode_plot(self.E.plot())
self.plot_link = '<img src="%s" width="200" height="150"/>' % self.plot
self.properties = [('Label', self.lmfdb_label),
(None, self.plot_link),
('Conductor', '\(%s\)' % data['conductor']),
('Discriminant', '\(%s\)' % data['disc']),
('j-invariant', '%s' % data['j_inv_latex']),
('CM', '%s' % data['CM']),
('Rank', '\(%s\)' % mw['rank']),
('Torsion Structure', '\(%s\)' % mw['tor_struct'])
]
self.title = "Elliptic Curve %s (Cremona label %s)" % (self.lmfdb_label, self.label)
self.bread = [('Elliptic Curves', url_for("ecnf.index")),
('$\Q$', 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_code_snippets(self):
# read in code.yaml from current directory:
_curdir = os.path.dirname(os.path.abspath(__file__))
self.code = yaml.load(open(os.path.join(_curdir, "code.yaml")))
# Fill in placeholders for this specific curve:
for lang in ['sage', 'pari', 'magma']:
self.code['curve'][lang] = self.code['curve'][lang] % (self.data['ainvs'],self.label)
for k in self.code:
if k != 'prompt':
for lang in self.code[k]:
self.code[k][lang] = self.code[k][lang].split("\n")
# remove final empty line
if len(self.code[k][lang][-1])==0:
self.code[k][lang] = self.code[k][lang][:-1]
def render_curve_webpage_by_label(label):
C = lmfdb.base.getDBConnection()
data = C.elliptic_curves.curves.find_one({'lmfdb_label': label})
if data is None:
return elliptic_curve_jump_error(label, {})
info = {}
ainvs = [int(a) for a in data['ainvs']]
E = EllipticCurve(ainvs)
cremona_label = data['label']
lmfdb_label = data['lmfdb_label']
N = ZZ(data['conductor'])
cremona_iso_class = data['iso'] # eg '37a'
lmfdb_iso_class = data['lmfdb_iso'] # eg '37.a'
rank = data['rank']
try:
j_invariant = QQ(str(data['jinv']))
except KeyError:
j_invariant = E.j_invariant()
if j_invariant == 0:
j_inv_factored = latex(0)
else:
j_inv_factored = latex(j_invariant.factor())
jinv = unicode(str(j_invariant))
CMD = 0
CM = "no"
EndE = "\(\Z\)"
if E.has_cm():
CMD = E.cm_discriminant()
CM = "yes (\(%s\))" % CMD
if CMD % 4 == 0:
d4 = ZZ(CMD) // 4
# r = d4.squarefree_part()
# f = (d4//r).isqrt()
# f="" if f==1 else str(f)
# EndE = "\(\Z[%s\sqrt{%s}]\)"%(f,r)
EndE = "\(\Z[\sqrt{%s}]\)" % (d4)
else:
EndE = "\(\Z[(1+\sqrt{%s})/2]\)" % CMD
# plot=E.plot()
discriminant = E.discriminant()
xintpoints_projective = [
E.lift_x(x)
for x in xintegral_point(data['x-coordinates_of_integral_points'])
]
xintpoints = proj_to_aff(xintpoints_projective)
if 'degree' in data:
modular_degree = data['degree']
else:
try:
modular_degree = E.modular_degree()
except RuntimeError:
modular_degree = 0 # invalid, will be displayed nicely
G = E.torsion_subgroup().gens()
E_pari = E.pari_curve(prec=200)
from sage.libs.pari.all import PariError
try:
minq = E.minimal_quadratic_twist()[0]
except PariError: # this does occur with 164411a1
print "PariError computing minimal quadratic twist of elliptic curve %s" % lmfdb_label
minq = E
if E == minq:
minq_label = lmfdb_label
else:
minq_ainvs = [str(c) for c in minq.ainvs()]
minq_label = C.elliptic_curves.curves.find_one({'ainvs': minq_ainvs
})['lmfdb_label']
# We do not just do the following, as Sage's installed database
# might not have all the curves in the LMFDB database.
# minq_label = E.minimal_quadratic_twist()[0].label()
if 'gens' in data:
generator = parse_gens(data['gens'])
if len(G) == 0:
tor_struct = '\mathrm{Trivial}'
tor_group = '\mathrm{Trivial}'
else:
tor_group = ' \\times '.join(['\Z/{%s}\Z' % a.order() for a in G])
if 'torsion_structure' in data:
info['tor_structure'] = ' \\times '.join(
['\Z/{%s}\Z' % int(a) for a in data['torsion_structure']])
else:
info['tor_structure'] = tor_group
def trim_galois_image_code(s):
return s[2:] if s[1].isdigit() else s[1:]
if 'galois_images' in data:
galois_images = data['galois_images']
galois_images = [trim_galois_image_code(s) for s in galois_images]
non_surjective_primes = data['non-surjective_primes']
galois_data = [{
'p': p,
'image': im
} for p, im in zip(non_surjective_primes, galois_images)]
info.update(data)
if rank >= 2:
lder_tex = "L%s(E,1)" % ("^{(" + str(rank) + ")}")
elif rank == 1:
lder_tex = "L%s(E,1)" % ("'" * rank)
else:
assert rank == 0
lder_tex = "L(E,1)"
info['Gamma0optimal'] = (cremona_label[-1] == '1'
if cremona_iso_class != '990h' else
cremona_label[-1] == '3')
info['modular_degree'] = modular_degree
p_adic_data_exists = (C.elliptic_curves.padic_db.find({
'lmfdb_iso':
lmfdb_iso_class
}).count()) > 0 and info['Gamma0optimal']
# Local data
local_data = []
for p in N.prime_factors():
local_info = E.local_data(p, algorithm="generic")
local_data.append({
'p':
p,
'tamagawa_number':
local_info.tamagawa_number(),
'kodaira_symbol':
web_latex(local_info.kodaira_symbol()).replace('$', ''),
'reduction_type':
local_info.bad_reduction_type()
})
mod_form_iso = lmfdb_label_regex.match(lmfdb_iso_class).groups()[1]
tamagawa_numbers = [
E.local_data(p, algorithm="generic").tamagawa_number()
for p in N.prime_factors()
]
# if we use E.tamagawa_numbers() it calls E.local_data(p) which
# crashes on some curves e.g. 164411a1
info.update({
'conductor':
N,
'disc_factor':
latex(discriminant.factor()),
'j_invar_factor':
j_inv_factored,
'label':
lmfdb_label,
'cremona_label':
cremona_label,
'iso_class':
lmfdb_iso_class,
'cremona_iso_class':
cremona_iso_class,
'equation':
web_latex(E),
#'f': ajax_more(E.q_eigenform, 10, 20, 50, 100, 250),
'f':
web_latex(E.q_eigenform(10)),
'generators':
', '.join(web_latex(g) for g in generator) if 'gens' in data else ' ',
'lder':
lder_tex,
'p_adic_primes': [
p for p in sage.all.prime_range(5, 100)
if E.is_ordinary(p) and not p.divides(N)
],
'p_adic_data_exists':
p_adic_data_exists,
'ainvs':
format_ainvs(data['ainvs']),
'CM':
CM,
'CMD':
CMD,
'EndE':
EndE,
'tamagawa_numbers':
r' \cdot '.join(str(sage.all.factor(c)) for c in tamagawa_numbers),
'local_data':
local_data,
'cond_factor':
latex(N.factor()),
'galois_data':
galois_data,
'xintegral_points':
', '.join(web_latex(P) for P in xintpoints),
'tor_gens':
', '.join(web_latex(eval(g))
for g in data['torsion_generators']) if False else ', '.join(
web_latex(P.element().xy()) for P in list(G))
})
info['friends'] = [('Isogeny class ' + lmfdb_iso_class,
url_for(".by_ec_label", label=lmfdb_iso_class)),
('Minimal quadratic twist ' + minq_label,
url_for(".by_ec_label", label=minq_label)),
('All twists ',
url_for(".rational_elliptic_curves", jinv=jinv)),
('L-function',
url_for("l_functions.l_function_ec_page",
label=lmfdb_label)),
('Symmetric square L-function',
url_for("l_functions.l_function_ec_sym_page",
power='2',
label=lmfdb_iso_class)),
('Symmetric 4th power L-function',
url_for("l_functions.l_function_ec_sym_page",
power='4',
label=lmfdb_iso_class))]
info['friends'].append(
('Modular form ' + lmfdb_iso_class.replace('.', '.2'),
url_for("emf.render_elliptic_modular_forms",
level=int(N),
weight=2,
character=0,
label=mod_form_iso)))
info['downloads'] = [('Download coeffients of q-expansion',
url_for(".download_EC_qexp",
label=lmfdb_label,
limit=100)),
('Download all stored data',
url_for(".download_EC_all", label=lmfdb_label))]
# info['learnmore'] = [('Elliptic Curves', url_for(".not_yet_implemented"))]
# info['plot'] = image_src(plot)
info['plot'] = url_for('.plot_ec', label=lmfdb_label)
properties2 = [('Label', '%s' % lmfdb_label),
(None, '<img src="%s" width="200" height="150"/>' %
url_for('.plot_ec', label=lmfdb_label)),
('Conductor', '\(%s\)' % N),
('Discriminant', '\(%s\)' % discriminant),
('j-invariant', '%s' % web_latex(j_invariant)),
('CM', '%s' % CM), ('Rank', '\(%s\)' % rank),
('Torsion Structure', '\(%s\)' % tor_group)]
# properties.extend([ "prop %s = %s<br/>" % (_,_*1923) for _ in range(12) ])
credit = 'John Cremona and Andrew Sutherland'
if info['label'] == info['cremona_label']:
t = "Elliptic Curve %s" % info['label']
else:
t = "Elliptic Curve %s (Cremona label %s)" % (info['label'],
info['cremona_label'])
bread = [('Elliptic Curves ', url_for(".rational_elliptic_curves")),
('Elliptic curves %s' % lmfdb_label, ' ')]
return render_template("curve.html",
properties2=properties2,
credit=credit,
bread=bread,
title=t,
info=info,
friends=info['friends'],
downloads=info['downloads'])
def render_curve_webpage_by_label(label):
C = base.getDBConnection()
data = C.ellcurves.curves.find_one({'label': label})
if data is None:
return "No such curve"
info = {}
ainvs = [int(a) for a in data['ainvs']]
E = EllipticCurve(ainvs)
label = data['label']
N = ZZ(data['conductor'])
iso_class = data['iso']
rank = data['rank']
j_invariant = E.j_invariant()
#plot=E.plot()
discriminant = E.discriminant()
xintpoints_projective = [
E.lift_x(x)
for x in xintegral_point(data['x-coordinates_of_integral_points'])
]
xintpoints = proj_to_aff(xintpoints_projective)
G = E.torsion_subgroup().gens()
if 'gens' in data:
generator = parse_gens(data['gens'])
if len(G) == 0:
tor_struct = 'Trivial'
tor_group = 'Trivial'
else:
tor_group = ' \\times '.join(
['\mathbb{Z}/{%s}\mathbb{Z}' % a.order() for a in G])
if 'torsion_structure' in data:
info['tor_structure'] = ' \\times '.join([
'\mathbb{Z}/{%s}\mathbb{Z}' % int(a)
for a in data['torsion_structure']
])
else:
info['tor_structure'] = tor_group
info.update(data)
if rank >= 2:
lder_tex = "L%s(E,1)" % ("^{(" + str(rank) + ")}")
elif rank == 1:
lder_tex = "L%s(E,1)" % ("'" * rank)
else:
assert rank == 0
lder_tex = "L(E,1)"
info.update({
'conductor':
N,
'disc_factor':
latex(discriminant.factor()),
'j_invar_factor':
latex(j_invariant.factor()),
'label':
label,
'isogeny':
iso_class,
'equation':
web_latex(E),
#'f': ajax_more(E.q_eigenform, 10, 20, 50, 100, 250),
'f':
web_latex(E.q_eigenform(10)),
'generators':
','.join(web_latex(g) for g in generator) if 'gens' in data else ' ',
'lder':
lder_tex,
'p_adic_primes': [
p for p in sage.all.prime_range(5, 100)
if E.is_ordinary(p) and not p.divides(N)
],
'ainvs':
format_ainvs(data['ainvs']),
'tamagawa_numbers':
r' \cdot '.join(str(sage.all.factor(c)) for c in E.tamagawa_numbers()),
'cond_factor':
latex(N.factor()),
'xintegral_points':
','.join(web_latex(i_p) for i_p in xintpoints),
'tor_gens':
','.join(web_latex(eval(g)) for g in data['torsion_generators'])
if 'torsion_generators' in data else list(G)
})
info['downloads_visible'] = True
info['downloads'] = [('worksheet', url_for("not_yet_implemented"))]
info['friends'] = [('Isogeny class', "/EllipticCurve/Q/%s" % iso_class),
('Modular Form',
url_for("emf.render_elliptic_modular_form_from_label",
label="%s" % (iso_class))),
('L-function', "/L/EllipticCurve/Q/%s" % label)]
info['learnmore'] = [('Elliptic Curves', url_for("not_yet_implemented"))]
#info['plot'] = image_src(plot)
info['plot'] = url_for('plot_ec', label=label)
info['iso_class'] = data['iso']
info['download_qexp_url'] = url_for('download_qexp',
limit=100,
ainvs=','.join([str(a)
for a in ainvs]))
properties2 = [('Label', '%s' % label),
(None, '<img src="%s" width="200" height="150"/>' %
url_for('plot_ec', label=label)),
('Conductor', '\(%s\)' % N),
('Discriminant', '\(%s\)' % discriminant),
('j-invariant', '\(%s\)' % j_invariant),
('Rank', '\(%s\)' % rank),
('Torsion Structure', '\(%s\)' % tor_group)]
#properties.extend([ "prop %s = %s<br/>" % (_,_*1923) for _ in range(12) ])
credit = 'John Cremona'
t = "Elliptic Curve %s" % info['label']
bread = [('Elliptic Curves ', url_for("rational_elliptic_curves")),
('Elliptic curves %s' % info['label'], ' ')]
return render_template("elliptic_curve/elliptic_curve.html",
info=info,
properties2=properties2,
credit=credit,
bread=bread,
title=t)
class WebEC(object):
"""
Class for an elliptic curve over Q
"""
def __init__(self, dbdata):
"""
Arguments:
- dbdata: the data from the database
"""
logger.info("Constructing an instance of ECisog_class")
self.__dict__.update(dbdata)
# Next lines because the hyphens make trouble
self.xintcoords = parse_list(dbdata['x-coordinates_of_integral_points'])
self.non_surjective_primes = dbdata['non-surjective_primes']
self.make_curve()
@staticmethod
def by_label(label):
"""
Searches for a specific elliptic curve in the curves
collection by its label, which can be either in LMFDB or
Cremona format.
"""
print "curve label = %s" % label
try:
N, iso, number = lmfdb_label_regex.match(label).groups()
data = db_ec().find_one({"lmfdb_label" : label})
except AttributeError:
try:
N, iso, number = cremona_label_regex.match(label).groups()
data = db_ec().find_one({"label" : label})
except AttributeError:
return "Invalid label" # caller must catch this and raise an error
if data:
return WebEC(data)
return "Curve not found" # caller must catch this and raise an error
def make_curve(self):
# To start with the data fields of self are just those from
# the database. We need to reformat these, construct the
# actual elliptic curve E, and compute some further (easy)
# data about it.
#
# Weierstrass equation
data = self.data = {}
data['ainvs'] = [int(ai) for ai in self.ainvs]
self.E = EllipticCurve(data['ainvs'])
data['equation'] = web_latex(self.E)
# conductor, j-invariant and discriminant
data['conductor'] = N = ZZ(self.conductor)
bad_primes = N.prime_factors()
try:
data['j_invariant'] = QQ(str(self.jinv))
except KeyError:
data['j_invariant'] = self.E.j_invariant()
data['j_inv_factor'] = latex(0)
if data['j_invariant']:
data['j_inv_factor'] = latex(data['j_invariant'].factor())
data['j_inv_str'] = unicode(str(data['j_invariant']))
data['j_inv_latex'] = web_latex(data['j_invariant'])
data['disc'] = self.E.discriminant()
data['disc_latex'] = web_latex(data['disc'])
data['disc_factor'] = latex(data['disc'].factor())
data['cond_factor'] =latex(N.factor())
data['cond_latex'] = web_latex(N)
# CM and endomorphism ring
data['CMD'] = 0
data['CM'] = "no"
data['EndE'] = "\(\Z\)"
if self.E.has_cm():
data['CMD'] = self.E.cm_discriminant()
data['CM'] = "yes (\(D=%s\))" % data['CMD']
if data['CMD']%4==0:
d4 = ZZ(data['CMD'])//4
data['EndE'] = "\(\Z[\sqrt{%s}]\)" % d4
else:
data['EndE'] = "\(\Z[(1+\sqrt{%s})/2]\)" % data['CMD']
# modular degree
try:
data['degree'] = self.degree
except AttributeError:
try:
data['degree'] = self.E.modular_degree()
except RuntimeError:
data['degree'] # invalid, but will be displayed nicely
# Minimal quadratic twist
E_pari = self.E.pari_curve(prec=200)
from sage.libs.pari.all import PariError
try:
minq = self.E.minimal_quadratic_twist()[0]
except PariError: # this does occur with 164411a1
ec.debug("PariError computing minimal quadratic twist of elliptic curve %s" % lmfdb_label)
minq = self.E
if self.E == minq:
data['minq_label'] = self.lmfdb_label
else:
minq_ainvs = [str(c) for c in minq.ainvs()]
data['minq_label'] = db_ec().find_one({'ainvs': minq_ainvs})['lmfdb_label']
# rational and integral points
mw = self.mw = {}
xintpoints_projective = [self.E.lift_x(x) for x in self.xintcoords]
xintpoints = [P.xy() for P in xintpoints_projective]
mw['int_points'] = ', '.join(web_latex(P) for P in xintpoints)
# Generators of infinite order
mw['rank'] = self.rank
try:
mw['generators'] = ', '.join(web_latex(self.E(g).xy()) for g in parse_points(self.gens))
except AttributeError:
mw['generators'] = ''
# Torsion subgroup: order, structure, generators
mw['tor_order'] = self.torsion
tor_struct = [int(c) for c in self.torsion_structure]
if mw['tor_order'] == 1:
mw['tor_struct'] = '\mathrm{Trivial}'
mw['tor_gens'] = ''
else:
mw['tor_struct'] = ' \\times '.join(['\Z/{%s}\Z' % n
for n in tor_struct])
mw['tor_gens'] = ', '.join(web_latex(self.E(g).xy()) for g in parse_points(self.torsion_generators))
# Images of Galois representations
try:
data['galois_images'] = [trim_galois_image_code(s) for s in self.galois_images]
data['non_surjective_primes'] = self.non_surjective_primes
except AttributeError:
print "No Galois image data"
data['galois_images'] = []
data['non_surjective_primes'] = []
data['galois_data'] = [{'p': p,'image': im }
for p,im in zip(data['non_surjective_primes'],
data['galois_images'])]
# Leading term of L-function & BSD data
bsd = self.bsd = {}
if mw['rank'] >= 2:
bsd['lder_name'] = "L^{(%s)}(E,1)" % mw['rank']
elif mw['rank']:
bsd['lder_name'] = "L'(E,1)"
else:
bsd['lder_name'] = "L(E,1)"
bsd['reg'] = self.regulator
bsd['omega'] = self.real_period
bsd['sha'] = int(0.1+self.sha_an)
bsd['lder'] = self.special_value
# 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)
if self.iso == '990h':
data['Gamma0optimal'] = bool(self.number == 3)
else:
data['Gamma0optimal'] = bool(self.number == 1)
data['p_adic_data_exists'] = False
if data['Gamma0optimal']:
data['p_adic_data_exists'] = (padic_db().find({'lmfdb_iso': self.lmfdb_iso}).count()) > 0
data['p_adic_primes'] = [p for p in sage.all.prime_range(5, 100)
if self.E.is_ordinary(p) and not p.divides(N)]
# Local data
local_data = self.local_data = []
# if we use E.tamagawa_numbers() it calls E.local_data(p) which
# crashes on some curves e.g. 164411a1
tamagawa_numbers = []
for p in bad_primes:
local_info = self.E.local_data(p, algorithm="generic")
local_data_p = {}
local_data_p['p'] = p
local_data_p['tamagawa_number'] = local_info.tamagawa_number()
tamagawa_numbers.append(ZZ(local_info.tamagawa_number()))
local_data_p['kodaira_symbol'] = web_latex(local_info.kodaira_symbol()).replace('$', '')
local_data_p['reduction_type'] = local_info.bad_reduction_type()
local_data.append(local_data_p)
bsd['tamagawa_factors'] = r' \cdot '.join(str(c.factor()) for c in tamagawa_numbers)
bsd['tamagawa_product'] = sage.misc.all.prod(tamagawa_numbers)
mod_form_iso = lmfdb_label_regex.match(self.lmfdb_iso).groups()[1]
data['newform'] = web_latex(self.E.q_eigenform(10))
self.friends = [
('Isogeny class ' + self.lmfdb_iso, url_for(".by_ec_label", label=self.lmfdb_iso)),
('Minimal quadratic twist ' + data['minq_label'], url_for(".by_ec_label", label=data['minq_label'])),
('All twists ', url_for(".rational_elliptic_curves", jinv=self.jinv)),
('L-function', url_for("l_functions.l_function_ec_page", label=self.lmfdb_label)),
('Symmetric square L-function', url_for("l_functions.l_function_ec_sym_page", power='2', label=self.lmfdb_iso)),
('Symmetric 4th power L-function', url_for("l_functions.l_function_ec_sym_page", power='4', label=self.lmfdb_iso)),
('Modular form ' + self.lmfdb_iso.replace('.', '.2'), url_for("emf.render_elliptic_modular_forms", level=int(N), weight=2, character=0, label=mod_form_iso))]
self.downloads = [('Download coeffients of q-expansion', url_for(".download_EC_qexp", label=self.lmfdb_label, limit=100)),
('Download all stored data', url_for(".download_EC_all", label=self.lmfdb_label))]
self.plot = encode_plot(self.E.plot())
self.plot_link = '<img src="%s" width="200" height="150"/>' % self.plot
self.properties = [('Label', self.lmfdb_label),
(None, self.plot_link),
('Conductor', '\(%s\)' % data['conductor']),
('Discriminant', '\(%s\)' % data['disc']),
('j-invariant', '%s' % data['j_inv_latex']),
('CM', '%s' % data['CM']),
('Rank', '\(%s\)' % mw['rank']),
('Torsion Structure', '\(%s\)' % mw['tor_struct'])
]
if self.lmfdb_iso == self.iso:
self.title = "Elliptic Curve %s" % self.lmfdb_label
else:
self.title = "Elliptic Curve %s (Cremona label %s)" % (self.lmfdb_label, self.label)
self.bread = [('Elliptic Curves ', url_for(".rational_elliptic_curves")), ('isogeny class %s' % self.lmfdb_iso, ' ')]
class WebEC(object):
"""
Class for an elliptic curve over Q
"""
def __init__(self, dbdata):
"""
Arguments:
- dbdata: the data from the database
"""
logger.debug("Constructing an instance of ECisog_class")
self.__dict__.update(dbdata)
# Next lines because the hyphens make trouble
self.xintcoords = split_list(dbdata["x-coordinates_of_integral_points"])
self.non_surjective_primes = dbdata["non-surjective_primes"]
# Next lines because the python identifiers cannot start with 2
self.twoadic_index = dbdata["2adic_index"]
self.twoadic_log_level = dbdata["2adic_log_level"]
self.twoadic_gens = dbdata["2adic_gens"]
self.twoadic_label = dbdata["2adic_label"]
# All other fields are handled here
self.make_curve()
@staticmethod
def by_label(label):
"""
Searches for a specific elliptic curve in the curves
collection by its label, which can be either in LMFDB or
Cremona format.
"""
try:
N, iso, number = split_lmfdb_label(label)
data = db_ec().find_one({"lmfdb_label": label})
except AttributeError:
try:
N, iso, number = split_cremona_label(label)
data = db_ec().find_one({"label": label})
except AttributeError:
return "Invalid label" # caller must catch this and raise an error
if data:
return WebEC(data)
return "Curve not found" # caller must catch this and raise an error
def make_curve(self):
# To start with the data fields of self are just those from
# the database. We need to reformat these, construct the
# actual elliptic curve E, and compute some further (easy)
# data about it.
#
# Weierstrass equation
data = self.data = {}
data["ainvs"] = [int(ai) for ai in self.ainvs]
self.E = EllipticCurve(data["ainvs"])
data["equation"] = web_latex(self.E)
# conductor, j-invariant and discriminant
data["conductor"] = N = ZZ(self.conductor)
bad_primes = N.prime_factors()
try:
data["j_invariant"] = QQ(str(self.jinv))
except KeyError:
data["j_invariant"] = self.E.j_invariant()
data["j_inv_factor"] = latex(0)
if data["j_invariant"]:
data["j_inv_factor"] = latex(data["j_invariant"].factor())
data["j_inv_str"] = unicode(str(data["j_invariant"]))
data["j_inv_latex"] = web_latex(data["j_invariant"])
data["disc"] = D = self.E.discriminant()
data["disc_latex"] = web_latex(data["disc"])
data["disc_factor"] = latex(data["disc"].factor())
data["cond_factor"] = latex(N.factor())
data["cond_latex"] = web_latex(N)
# CM and endomorphism ring
data["CMD"] = self.cm
data["CM"] = "no"
data["EndE"] = "\(\Z\)"
if self.cm:
data["CM"] = "yes (\(D=%s\))" % data["CMD"]
if data["CMD"] % 4 == 0:
d4 = ZZ(data["CMD"]) // 4
data["EndE"] = "\(\Z[\sqrt{%s}]\)" % d4
else:
data["EndE"] = "\(\Z[(1+\sqrt{%s})/2]\)" % data["CMD"]
data["ST"] = '<a href="%s">$%s$</a>' % (url_for("st.by_label", label="1.2.N(U(1))"), "N(\\mathrm{U}(1))")
else:
data["ST"] = '<a href="%s">$%s$</a>' % (url_for("st.by_label", label="1.2.SU(2)"), "\\mathrm{SU}(2)")
# modular degree
try:
data["degree"] = self.degree
except AttributeError:
try:
data["degree"] = self.E.modular_degree()
except RuntimeError:
data["degree"] # invalid, but will be displayed nicely
# Minimal quadratic twist
E_pari = self.E.pari_curve()
from sage.libs.pari.all import PariError
try:
minq, minqD = self.E.minimal_quadratic_twist()
except PariError: # this does occur with 164411a1
ec.debug("PariError computing minimal quadratic twist of elliptic curve %s" % lmfdb_label)
minq = self.E
minqD = 1
data["minq_D"] = minqD
if self.E == minq:
data["minq_label"] = self.lmfdb_label
data["minq_info"] = "(itself)"
else:
minq_ainvs = [str(c) for c in minq.ainvs()]
data["minq_label"] = db_ec().find_one({"jinv": str(self.E.j_invariant()), "ainvs": minq_ainvs})[
"lmfdb_label"
]
data["minq_info"] = "(by %s)" % minqD
minq_N, minq_iso, minq_number = split_lmfdb_label(data["minq_label"])
# rational and integral points
mw = self.mw = {}
xintpoints_projective = [self.E.lift_x(x) for x in self.xintcoords]
xintpoints = [P.xy() for P in xintpoints_projective]
mw["int_points"] = ", ".join(web_latex(P) for P in xintpoints)
# Generators of infinite order
mw["rank"] = self.rank
try:
self.generators = [self.E(g) for g in parse_points(self.gens)]
mw["generators"] = [web_latex(P.xy()) for P in self.generators]
mw["heights"] = [P.height() for P in self.generators]
except AttributeError:
mw["generators"] = ""
mw["heights"] = []
# Torsion subgroup: order, structure, generators
mw["tor_order"] = self.torsion
tor_struct = [int(c) for c in self.torsion_structure]
if mw["tor_order"] == 1:
mw["tor_struct"] = "\mathrm{Trivial}"
mw["tor_gens"] = ""
else:
mw["tor_struct"] = " \\times ".join(["\Z/{%s}\Z" % n for n in tor_struct])
mw["tor_gens"] = ", ".join(web_latex(self.E(g).xy()) for g in parse_points(self.torsion_generators))
# Images of Galois representations
try:
data["galois_images"] = [trim_galois_image_code(s) for s in self.galois_images]
data["non_surjective_primes"] = self.non_surjective_primes
except AttributeError:
# print "No Galois image data"
data["galois_images"] = []
data["non_surjective_primes"] = []
data["galois_data"] = [
{"p": p, "image": im} for p, im in zip(data["non_surjective_primes"], data["galois_images"])
]
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 & BSD data
bsd = self.bsd = {}
r = self.rank
if r >= 2:
bsd["lder_name"] = "L^{(%s)}(E,1)/%s!" % (r, r)
elif r:
bsd["lder_name"] = "L'(E,1)"
else:
bsd["lder_name"] = "L(E,1)"
bsd["reg"] = self.regulator
bsd["omega"] = self.real_period
bsd["sha"] = int(0.1 + self.sha_an)
bsd["lder"] = self.special_value
# 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)
if self.iso == "990h":
data["Gamma0optimal"] = bool(self.number == 3)
else:
data["Gamma0optimal"] = bool(self.number == 1)
data["p_adic_data_exists"] = False
if data["Gamma0optimal"]:
data["p_adic_data_exists"] = (padic_db().find({"lmfdb_iso": self.lmfdb_iso}).count()) > 0
data["p_adic_primes"] = [p for p in sage.all.prime_range(5, 100) if self.E.is_ordinary(p) and not p.divides(N)]
# Local data
local_data = self.local_data = []
# if we use E.tamagawa_numbers() it calls E.local_data(p) which
# used to crash on some curves e.g. 164411a1
tamagawa_numbers = []
for p in bad_primes:
local_info = self.E.local_data(p, algorithm="generic")
local_data_p = {}
local_data_p["p"] = p
local_data_p["tamagawa_number"] = local_info.tamagawa_number()
tamagawa_numbers.append(ZZ(local_info.tamagawa_number()))
local_data_p["kodaira_symbol"] = web_latex(local_info.kodaira_symbol()).replace("$", "")
local_data_p["reduction_type"] = local_info.bad_reduction_type()
local_data_p["ord_cond"] = local_info.conductor_valuation()
local_data_p["ord_disc"] = local_info.discriminant_valuation()
local_data_p["ord_den_j"] = max(0, -self.E.j_invariant().valuation(p))
local_data.append(local_data_p)
cp_fac = [cp.factor() for cp in tamagawa_numbers]
cp_fac = [latex(cp) if len(cp) < 2 else "(" + latex(cp) + ")" for cp in cp_fac]
bsd["tamagawa_factors"] = r"\cdot".join(cp_fac)
bsd["tamagawa_product"] = sage.misc.all.prod(tamagawa_numbers)
cond, iso, num = split_lmfdb_label(self.lmfdb_label)
data["newform"] = web_latex(self.E.q_eigenform(10))
self.newform_label = newform_label(cond, 2, 1, iso)
self.newform_link = url_for("emf.render_elliptic_modular_forms", level=cond, weight=2, character=1, label=iso)
newform_exists_in_db = is_newform_in_db(self.newform_label)
self._code = None
self.friends = [
("Isogeny class " + self.lmfdb_iso, url_for(".by_double_iso_label", conductor=N, iso_label=iso)),
(
"Minimal quadratic twist %s %s" % (data["minq_info"], data["minq_label"]),
url_for(".by_triple_label", conductor=minq_N, iso_label=minq_iso, number=minq_number),
),
("All twists ", url_for(".rational_elliptic_curves", jinv=self.jinv)),
("L-function", url_for("l_functions.l_function_ec_page", label=self.lmfdb_label)),
]
if not self.cm:
if N <= 300:
self.friends += [
(
"Symmetric square L-function",
url_for("l_functions.l_function_ec_sym_page", power="2", label=self.lmfdb_iso),
)
]
if N <= 50:
self.friends += [
(
"Symmetric cube L-function",
url_for("l_functions.l_function_ec_sym_page", power="3", label=self.lmfdb_iso),
)
]
if newform_exists_in_db:
self.friends += [("Modular form " + self.newform_label, self.newform_link)]
self.downloads = [
("Download coefficients of q-expansion", url_for(".download_EC_qexp", label=self.lmfdb_label, limit=100)),
("Download all stored data", url_for(".download_EC_all", label=self.lmfdb_label)),
(
"Download Magma code",
url_for(
".ec_code_download",
conductor=cond,
iso=iso,
number=num,
label=self.lmfdb_label,
download_type="magma",
),
),
(
"Download Sage code",
url_for(
".ec_code_download",
conductor=cond,
iso=iso,
number=num,
label=self.lmfdb_label,
download_type="sage",
),
),
(
"Download GP code",
url_for(
".ec_code_download", conductor=cond, iso=iso, number=num, label=self.lmfdb_label, download_type="gp"
),
),
]
self.plot = encode_plot(self.E.plot())
self.plot_link = '<img src="%s" width="200" height="150"/>' % self.plot
self.properties = [
("Label", self.lmfdb_label),
(None, self.plot_link),
("Conductor", "\(%s\)" % data["conductor"]),
("Discriminant", "\(%s\)" % data["disc"]),
("j-invariant", "%s" % data["j_inv_latex"]),
("CM", "%s" % data["CM"]),
("Rank", "\(%s\)" % mw["rank"]),
("Torsion Structure", "\(%s\)" % mw["tor_struct"]),
]
self.title = "Elliptic Curve %s (Cremona label %s)" % (self.lmfdb_label, self.label)
self.bread = [
("Elliptic Curves", url_for("ecnf.index")),
("$\Q$", 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 code(self):
if self._code == None:
self.make_code_snippets()
return self._code
def make_code_snippets(self):
# read in code.yaml from current directory:
_curdir = os.path.dirname(os.path.abspath(__file__))
self._code = yaml.load(open(os.path.join(_curdir, "code.yaml")))
# Fill in placeholders for this specific curve:
for lang in ["sage", "pari", "magma"]:
self._code["curve"][lang] = self._code["curve"][lang] % (self.data["ainvs"], self.label)
for k in self._code:
if k != "prompt":
for lang in self._code[k]:
self._code[k][lang] = self._code[k][lang].split("\n")
# remove final empty line
if len(self._code[k][lang][-1]) == 0:
self._code[k][lang] = self._code[k][lang][:-1]
class WebEC(object):
"""
Class for an elliptic curve over Q
"""
def __init__(self, dbdata):
"""
Arguments:
- dbdata: the data from the database
"""
logger.debug("Constructing an instance of ECisog_class")
self.__dict__.update(dbdata)
# Next lines because the hyphens make trouble
self.xintcoords = parse_list(
dbdata['x-coordinates_of_integral_points'])
self.non_surjective_primes = dbdata['non-surjective_primes']
# Next lines because the python identifiers cannot start with 2
self.twoadic_index = dbdata['2adic_index']
self.twoadic_log_level = dbdata['2adic_log_level']
self.twoadic_gens = dbdata['2adic_gens']
self.twoadic_label = dbdata['2adic_label']
# All other fields are handled here
self.make_curve()
@staticmethod
def by_label(label):
"""
Searches for a specific elliptic curve in the curves
collection by its label, which can be either in LMFDB or
Cremona format.
"""
try:
N, iso, number = split_lmfdb_label(label)
data = db_ec().find_one({"lmfdb_label": label})
except AttributeError:
try:
N, iso, number = split_cremona_label(label)
data = db_ec().find_one({"label": label})
except AttributeError:
return "Invalid label" # caller must catch this and raise an error
if data:
return WebEC(data)
return "Curve not found" # caller must catch this and raise an error
def make_curve(self):
# To start with the data fields of self are just those from
# the database. We need to reformat these, construct the
# actual elliptic curve E, and compute some further (easy)
# data about it.
#
# Weierstrass equation
data = self.data = {}
data['ainvs'] = [int(ai) for ai in self.ainvs]
self.E = EllipticCurve(data['ainvs'])
data['equation'] = web_latex(self.E)
# conductor, j-invariant and discriminant
data['conductor'] = N = ZZ(self.conductor)
bad_primes = N.prime_factors()
try:
data['j_invariant'] = QQ(str(self.jinv))
except KeyError:
data['j_invariant'] = self.E.j_invariant()
data['j_inv_factor'] = latex(0)
if data['j_invariant']:
data['j_inv_factor'] = latex(data['j_invariant'].factor())
data['j_inv_str'] = unicode(str(data['j_invariant']))
data['j_inv_latex'] = web_latex(data['j_invariant'])
data['disc'] = D = self.E.discriminant()
data['disc_latex'] = web_latex(data['disc'])
data['disc_factor'] = latex(data['disc'].factor())
data['cond_factor'] = latex(N.factor())
data['cond_latex'] = web_latex(N)
# CM and endomorphism ring
data['CMD'] = self.cm
data['CM'] = "no"
data['EndE'] = "\(\Z\)"
if self.cm:
data['CM'] = "yes (\(D=%s\))" % data['CMD']
if data['CMD'] % 4 == 0:
d4 = ZZ(data['CMD']) // 4
data['EndE'] = "\(\Z[\sqrt{%s}]\)" % d4
else:
data['EndE'] = "\(\Z[(1+\sqrt{%s})/2]\)" % data['CMD']
# modular degree
try:
data['degree'] = self.degree
except AttributeError:
try:
data['degree'] = self.E.modular_degree()
except RuntimeError:
data['degree'] # invalid, but will be displayed nicely
# Minimal quadratic twist
E_pari = self.E.pari_curve()
from sage.libs.pari.all import PariError
try:
minq, minqD = self.E.minimal_quadratic_twist()
except PariError: # this does occur with 164411a1
ec.debug(
"PariError computing minimal quadratic twist of elliptic curve %s"
% lmfdb_label)
minq = self.E
minqD = 1
data['minq_D'] = minqD
if self.E == minq:
data['minq_label'] = self.lmfdb_label
data['minq_info'] = '(itself)'
else:
minq_ainvs = [str(c) for c in minq.ainvs()]
data['minq_label'] = db_ec().find_one({'ainvs':
minq_ainvs})['lmfdb_label']
data['minq_info'] = '(by %s)' % minqD
minq_N, minq_iso, minq_number = split_lmfdb_label(data['minq_label'])
# rational and integral points
mw = self.mw = {}
xintpoints_projective = [self.E.lift_x(x) for x in self.xintcoords]
xintpoints = [P.xy() for P in xintpoints_projective]
mw['int_points'] = ', '.join(web_latex(P) for P in xintpoints)
# Generators of infinite order
mw['rank'] = self.rank
try:
self.generators = [self.E(g) for g in parse_points(self.gens)]
mw['generators'] = [web_latex(P.xy()) for P in self.generators]
mw['heights'] = [P.height() for P in self.generators]
except AttributeError:
mw['generators'] = ''
mw['heights'] = []
# Torsion subgroup: order, structure, generators
mw['tor_order'] = self.torsion
tor_struct = [int(c) for c in self.torsion_structure]
if mw['tor_order'] == 1:
mw['tor_struct'] = '\mathrm{Trivial}'
mw['tor_gens'] = ''
else:
mw['tor_struct'] = ' \\times '.join(
['\Z/{%s}\Z' % n for n in tor_struct])
mw['tor_gens'] = ', '.join(
web_latex(self.E(g).xy())
for g in parse_points(self.torsion_generators))
# Images of Galois representations
try:
data['galois_images'] = [
trim_galois_image_code(s) for s in self.galois_images
]
data['non_surjective_primes'] = self.non_surjective_primes
except AttributeError:
#print "No Galois image data"
data['galois_images'] = []
data['non_surjective_primes'] = []
data['galois_data'] = [{
'p': p,
'image': im
} for p, im in zip(data['non_surjective_primes'],
data['galois_images'])]
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 & BSD data
bsd = self.bsd = {}
r = self.rank
if r >= 2:
bsd['lder_name'] = "L^{(%s)}(E,1)/%s!" % (r, r)
elif r:
bsd['lder_name'] = "L'(E,1)"
else:
bsd['lder_name'] = "L(E,1)"
bsd['reg'] = self.regulator
bsd['omega'] = self.real_period
bsd['sha'] = int(0.1 + self.sha_an)
bsd['lder'] = self.special_value
# 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)
if self.iso == '990h':
data['Gamma0optimal'] = bool(self.number == 3)
else:
data['Gamma0optimal'] = bool(self.number == 1)
data['p_adic_data_exists'] = False
if data['Gamma0optimal']:
data['p_adic_data_exists'] = (padic_db().find({
'lmfdb_iso':
self.lmfdb_iso
}).count()) > 0
data['p_adic_primes'] = [
p for p in sage.all.prime_range(5, 100)
if self.E.is_ordinary(p) and not p.divides(N)
]
# Local data
local_data = self.local_data = []
# if we use E.tamagawa_numbers() it calls E.local_data(p) which
# used to crash on some curves e.g. 164411a1
tamagawa_numbers = []
for p in bad_primes:
local_info = self.E.local_data(p, algorithm="generic")
local_data_p = {}
local_data_p['p'] = p
local_data_p['tamagawa_number'] = local_info.tamagawa_number()
tamagawa_numbers.append(ZZ(local_info.tamagawa_number()))
local_data_p['kodaira_symbol'] = web_latex(
local_info.kodaira_symbol()).replace('$', '')
local_data_p['reduction_type'] = local_info.bad_reduction_type()
local_data_p['ord_cond'] = local_info.conductor_valuation()
local_data_p['ord_disc'] = local_info.discriminant_valuation()
local_data_p['ord_den_j'] = max(0,
-self.E.j_invariant().valuation(p))
local_data.append(local_data_p)
cp_fac = [cp.factor() for cp in tamagawa_numbers]
cp_fac = [
latex(cp) if len(cp) < 2 else '(' + latex(cp) + ')'
for cp in cp_fac
]
bsd['tamagawa_factors'] = r'\cdot'.join(cp_fac)
bsd['tamagawa_product'] = sage.misc.all.prod(tamagawa_numbers)
cond, iso, num = split_lmfdb_label(self.lmfdb_label)
data['newform'] = web_latex(self.E.q_eigenform(10))
self.make_code_snippets()
self.friends = [('Isogeny class ' + self.lmfdb_iso,
url_for(".by_double_iso_label",
conductor=N,
iso_label=iso)),
('Minimal quadratic twist %s %s' %
(data['minq_info'], data['minq_label']),
url_for(".by_triple_label",
conductor=minq_N,
iso_label=minq_iso,
number=minq_number)),
('All twists ',
url_for(".rational_elliptic_curves", jinv=self.jinv)),
('L-function',
url_for("l_functions.l_function_ec_page",
label=self.lmfdb_label)),
('Symmetric square L-function',
url_for("l_functions.l_function_ec_sym_page",
power='2',
label=self.lmfdb_iso)),
('Symmetric 4th power L-function',
url_for("l_functions.l_function_ec_sym_page",
power='4',
label=self.lmfdb_iso)),
('Modular form ' + self.lmfdb_iso.replace('.', '.2'),
url_for("emf.render_elliptic_modular_forms",
level=int(N),
weight=2,
character=1,
label=iso))]
self.downloads = [('Download coefficients of q-expansion',
url_for(".download_EC_qexp",
label=self.lmfdb_label,
limit=100)),
('Download all stored data',
url_for(".download_EC_all",
label=self.lmfdb_label))]
self.plot = encode_plot(self.E.plot())
self.plot_link = '<img src="%s" width="200" height="150"/>' % self.plot
self.properties = [('Label', self.lmfdb_label), (None, self.plot_link),
('Conductor', '\(%s\)' % data['conductor']),
('Discriminant', '\(%s\)' % data['disc']),
('j-invariant', '%s' % data['j_inv_latex']),
('CM', '%s' % data['CM']),
('Rank', '\(%s\)' % mw['rank']),
('Torsion Structure', '\(%s\)' % mw['tor_struct'])]
self.title = "Elliptic Curve %s (Cremona label %s)" % (
self.lmfdb_label, self.label)
self.bread = [('Elliptic Curves', url_for("ecnf.index")),
('$\Q$', 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_code_snippets(self):
# read in code.yaml from current directory:
_curdir = os.path.dirname(os.path.abspath(__file__))
self.code = yaml.load(open(os.path.join(_curdir, "code.yaml")))
# Fill in placeholders for this specific curve:
for lang in ['sage', 'pari', 'magma']:
self.code['curve'][lang] = self.code['curve'][lang] % (
self.data['ainvs'], self.label)
for k in self.code:
if k != 'prompt':
for lang in self.code[k]:
self.code[k][lang] = self.code[k][lang].split("\n")
# remove final empty line
if len(self.code[k][lang][-1]) == 0:
self.code[k][lang] = self.code[k][lang][:-1]
def render_curve_webpage_by_label(label):
C = lmfdb.base.getDBConnection()
data = C.elliptic_curves.curves.find_one({'lmfdb_label': label})
if data is None:
return elliptic_curve_jump_error(label, {})
info = {}
ainvs = [int(a) for a in data['ainvs']]
E = EllipticCurve(ainvs)
cremona_label = data['label']
lmfdb_label = data['lmfdb_label']
N = ZZ(data['conductor'])
cremona_iso_class = data['iso'] # eg '37a'
lmfdb_iso_class = data['lmfdb_iso'] # eg '37.a'
rank = data['rank']
try:
j_invariant = QQ(str(data['jinv']))
except KeyError:
j_invariant = E.j_invariant()
if j_invariant == 0:
j_inv_factored = latex(0)
else:
j_inv_factored = latex(j_invariant.factor())
jinv = unicode(str(j_invariant))
CMD = 0
CM = "no"
EndE = "\(\Z\)"
if E.has_cm():
CMD = E.cm_discriminant()
CM = "yes (\(%s\))"%CMD
if CMD%4==0:
d4 = ZZ(CMD)//4
# r = d4.squarefree_part()
# f = (d4//r).isqrt()
# f="" if f==1 else str(f)
# EndE = "\(\Z[%s\sqrt{%s}]\)"%(f,r)
EndE = "\(\Z[\sqrt{%s}]\)"%(d4)
else:
EndE = "\(\Z[(1+\sqrt{%s})/2]\)"%CMD
# plot=E.plot()
discriminant = E.discriminant()
xintpoints_projective = [E.lift_x(x) for x in xintegral_point(data['x-coordinates_of_integral_points'])]
xintpoints = proj_to_aff(xintpoints_projective)
if 'degree' in data:
modular_degree = data['degree']
else:
try:
modular_degree = E.modular_degree()
except RuntimeError:
modular_degree = 0 # invalid, will be displayed nicely
G = E.torsion_subgroup().gens()
minq = E.minimal_quadratic_twist()[0]
if E == minq:
minq_label = lmfdb_label
else:
minq_ainvs = [str(c) for c in minq.ainvs()]
minq_label = C.elliptic_curves.curves.find_one({'ainvs': minq_ainvs})['lmfdb_label']
# We do not just do the following, as Sage's installed database
# might not have all the curves in the LMFDB database.
# minq_label = E.minimal_quadratic_twist()[0].label()
if 'gens' in data:
generator = parse_gens(data['gens'])
if len(G) == 0:
tor_struct = '\mathrm{Trivial}'
tor_group = '\mathrm{Trivial}'
else:
tor_group = ' \\times '.join(['\Z/{%s}\Z' % a.order() for a in G])
if 'torsion_structure' in data:
info['tor_structure'] = ' \\times '.join(['\Z/{%s}\Z' % int(a) for a in data['torsion_structure']])
else:
info['tor_structure'] = tor_group
info.update(data)
if rank >= 2:
lder_tex = "L%s(E,1)" % ("^{(" + str(rank) + ")}")
elif rank == 1:
lder_tex = "L%s(E,1)" % ("'" * rank)
else:
assert rank == 0
lder_tex = "L(E,1)"
info['Gamma0optimal'] = (
cremona_label[-1] == '1' if cremona_iso_class != '990h' else cremona_label[-1] == '3')
info['modular_degree'] = modular_degree
p_adic_data_exists = (C.elliptic_curves.padic_db.find(
{'lmfdb_iso': lmfdb_iso_class}).count()) > 0 and info['Gamma0optimal']
# Local data
local_data = []
for p in N.prime_factors():
local_info = E.local_data(p)
local_data.append({'p': p,
'tamagawa_number': local_info.tamagawa_number(),
'kodaira_symbol': web_latex(local_info.kodaira_symbol()).replace('$', ''),
'reduction_type': local_info.bad_reduction_type()
})
mod_form_iso = lmfdb_label_regex.match(lmfdb_iso_class).groups()[1]
info.update({
'conductor': N,
'disc_factor': latex(discriminant.factor()),
'j_invar_factor': j_inv_factored,
'label': lmfdb_label,
'cremona_label': cremona_label,
'iso_class': lmfdb_iso_class,
'cremona_iso_class': cremona_iso_class,
'equation': web_latex(E),
#'f': ajax_more(E.q_eigenform, 10, 20, 50, 100, 250),
'f': web_latex(E.q_eigenform(10)),
'generators': ', '.join(web_latex(g) for g in generator) if 'gens' in data else ' ',
'lder': lder_tex,
'p_adic_primes': [p for p in sage.all.prime_range(5, 100) if E.is_ordinary(p) and not p.divides(N)],
'p_adic_data_exists': p_adic_data_exists,
'ainvs': format_ainvs(data['ainvs']),
'CM': CM,
'CMD': CMD,
'EndE': EndE,
'tamagawa_numbers': r' \cdot '.join(str(sage.all.factor(c)) for c in E.tamagawa_numbers()),
'local_data': local_data,
'cond_factor': latex(N.factor()),
'xintegral_points': ', '.join(web_latex(P) for P in xintpoints),
'tor_gens': ', '.join(web_latex(eval(g)) for g in data['torsion_generators']) if False else ', '.join(web_latex(P.element().xy()) for P in list(G))
})
info['friends'] = [
('Isogeny class ' + lmfdb_iso_class, "/EllipticCurve/Q/%s" % lmfdb_iso_class),
('Minimal quadratic twist ' + minq_label, "/EllipticCurve/Q/%s" % minq_label),
('All twists ', url_for("rational_elliptic_curves", jinv=jinv)),
('L-function', url_for("l_functions.l_function_ec_page", label=lmfdb_label)),
('Symmetric square L-function', url_for("l_functions.l_function_ec_sym_page", power='2',
label=lmfdb_iso_class)),
('Symmetric 4th power L-function', url_for("l_functions.l_function_ec_sym_page", power='4',
label=lmfdb_iso_class))]
info['friends'].append(('Modular form ' + lmfdb_iso_class.replace('.', '.2'), url_for(
"emf.render_elliptic_modular_forms", level=int(N), weight=2, character=0, label=mod_form_iso)))
info['downloads'] = [('Download coeffients of q-expansion', url_for("download_EC_qexp", label=lmfdb_label, limit=100)),
('Download all stored data', url_for("download_EC_all", label=lmfdb_label))]
# info['learnmore'] = [('Elliptic Curves', url_for("not_yet_implemented"))]
# info['plot'] = image_src(plot)
info['plot'] = url_for('plot_ec', label=lmfdb_label)
properties2 = [('Label', '%s' % lmfdb_label),
(None, '<img src="%s" width="200" height="150"/>' % url_for(
'plot_ec', label=lmfdb_label)),
('Conductor', '\(%s\)' % N),
('Discriminant', '\(%s\)' % discriminant),
('j-invariant', '%s' % web_latex(j_invariant)),
('CM', '%s' % CM),
('Rank', '\(%s\)' % rank),
('Torsion Structure', '\(%s\)' % tor_group)
]
# properties.extend([ "prop %s = %s<br/>" % (_,_*1923) for _ in range(12) ])
credit = 'John Cremona'
if info['label'] == info['cremona_label']:
t = "Elliptic Curve %s" % info['label']
else:
t = "Elliptic Curve %s (Cremona label %s)" % (info['label'], info['cremona_label'])
bread = [('Elliptic Curves ', url_for("rational_elliptic_curves")), ('Elliptic curves %s' %
lmfdb_label, ' ')]
return render_template("elliptic_curve/elliptic_curve.html",
properties2=properties2, credit=credit, bread=bread, title=t, info=info, friends=info['friends'], downloads=info['downloads'])
def render_curve_webpage_by_label(label):
C = base.getDBConnection()
data = C.ellcurves.curves.find_one({'label': label})
if data is None:
return "No such curve"
info = {}
ainvs = [int(a) for a in data['ainvs']]
E = EllipticCurve(ainvs)
label=data['label']
N = ZZ(data['conductor'])
iso_class = data['iso']
rank = data['rank']
j_invariant=E.j_invariant()
#plot=E.plot()
discriminant=E.discriminant()
xintpoints_projective=[E.lift_x(x) for x in xintegral_point(data['x-coordinates_of_integral_points'])]
xintpoints=proj_to_aff(xintpoints_projective)
G = E.torsion_subgroup().gens()
if 'gens' in data:
generator=parse_gens(data['gens'])
if len(G) == 0:
tor_struct = 'Trivial'
tor_group='Trivial'
else:
tor_group=' \\times '.join(['\mathbb{Z}/{%s}\mathbb{Z}'%a.order() for a in G])
if 'torsion_structure' in data:
info['tor_structure']= ' \\times '.join(['\mathbb{Z}/{%s}\mathbb{Z}'% int(a) for a in data['torsion_structure']])
else:
info['tor_structure'] = tor_group
info.update(data)
if rank >=2:
lder_tex = "L%s(E,1)" % ("^{("+str(rank)+")}")
elif rank ==1:
lder_tex = "L%s(E,1)" % ("'"*rank)
else:
assert rank == 0
lder_tex = "L(E,1)"
info.update({
'conductor': N,
'disc_factor': latex(discriminant.factor()),
'j_invar_factor':latex(j_invariant.factor()),
'label': label,
'isogeny':iso_class,
'equation': web_latex(E),
#'f': ajax_more(E.q_eigenform, 10, 20, 50, 100, 250),
'f' : web_latex(E.q_eigenform(10)),
'generators':','.join(web_latex(g) for g in generator) if 'gens' in data else ' ',
'lder' : lder_tex,
'p_adic_primes': [p for p in sage.all.prime_range(5,100) if E.is_ordinary(p) and not p.divides(N)],
'ainvs': format_ainvs(data['ainvs']),
'tamagawa_numbers': r' \cdot '.join(str(sage.all.factor(c)) for c in E.tamagawa_numbers()),
'cond_factor':latex(N.factor()),
'xintegral_points':','.join(web_latex(i_p) for i_p in xintpoints),
'tor_gens':','.join(web_latex(eval(g)) for g in data['torsion_generators']) if 'torsion_generators' in data else list(G)
})
info['downloads_visible'] = True
info['downloads'] = [('worksheet', url_for("not_yet_implemented"))]
info['friends'] = [('Isogeny class', "/EllipticCurve/Q/%s" % iso_class),
('Modular Form', url_for("emf.render_elliptic_modular_form_from_label",label="%s" %(iso_class))),
('L-function', "/L/EllipticCurve/Q/%s" % label)]
info['learnmore'] = [('Elliptic Curves', url_for("not_yet_implemented"))]
#info['plot'] = image_src(plot)
info['plot'] = url_for('plot_ec', label=label)
info['iso_class'] = data['iso']
info['download_qexp_url'] = url_for('download_qexp', limit=100, ainvs=','.join([str(a) for a in ainvs]))
properties2 = [('Label', '%s' % label),
(None, '<img src="%s" width="200" height="150"/>' % url_for('plot_ec', label=label) ),
('Conductor', '\(%s\)' % N),
('Discriminant', '\(%s\)' % discriminant),
('j-invariant', '\(%s\)' % j_invariant),
('Rank', '\(%s\)' % rank),
('Torsion Structure', '\(%s\)' % tor_group)
]
#properties.extend([ "prop %s = %s<br/>" % (_,_*1923) for _ in range(12) ])
credit = 'John Cremona'
t = "Elliptic Curve %s" % info['label']
bread = [('Elliptic Curves ', url_for("rational_elliptic_curves")),('Elliptic curves %s' %info['label'],' ')]
return render_template("elliptic_curve/elliptic_curve.html",
info=info, properties2=properties2, credit=credit,bread=bread, title = t)
def render_curve_webpage_by_label(label):
C = lmfdb.base.getDBConnection()
data = C.elliptic_curves.curves.find_one({"lmfdb_label": label})
if data is None:
return elliptic_curve_jump_error(label, {})
info = {}
ainvs = [int(a) for a in data["ainvs"]]
E = EllipticCurve(ainvs)
cremona_label = data["label"]
lmfdb_label = data["lmfdb_label"]
N = ZZ(data["conductor"])
cremona_iso_class = data["iso"] # eg '37a'
lmfdb_iso_class = data["lmfdb_iso"] # eg '37.a'
rank = data["rank"]
try:
j_invariant = QQ(str(data["jinv"]))
except KeyError:
j_invariant = E.j_invariant()
if j_invariant == 0:
j_inv_factored = latex(0)
else:
j_inv_factored = latex(j_invariant.factor())
jinv = unicode(str(j_invariant))
CMD = 0
CM = "no"
EndE = "\(\Z\)"
if E.has_cm():
CMD = E.cm_discriminant()
CM = "yes (\(%s\))" % CMD
if CMD % 4 == 0:
d4 = ZZ(CMD) // 4
# r = d4.squarefree_part()
# f = (d4//r).isqrt()
# f="" if f==1 else str(f)
# EndE = "\(\Z[%s\sqrt{%s}]\)"%(f,r)
EndE = "\(\Z[\sqrt{%s}]\)" % (d4)
else:
EndE = "\(\Z[(1+\sqrt{%s})/2]\)" % CMD
# plot=E.plot()
discriminant = E.discriminant()
xintpoints_projective = [E.lift_x(x) for x in xintegral_point(data["x-coordinates_of_integral_points"])]
xintpoints = proj_to_aff(xintpoints_projective)
if "degree" in data:
modular_degree = data["degree"]
else:
try:
modular_degree = E.modular_degree()
except RuntimeError:
modular_degree = 0 # invalid, will be displayed nicely
G = E.torsion_subgroup().gens()
E_pari = E.pari_curve(prec=200)
from sage.libs.pari.all import PariError
try:
minq = E.minimal_quadratic_twist()[0]
except PariError: # this does occur with 164411a1
print "PariError computing minimal quadratic twist of elliptic curve %s" % lmfdb_label
minq = E
if E == minq:
minq_label = lmfdb_label
else:
minq_ainvs = [str(c) for c in minq.ainvs()]
minq_label = C.elliptic_curves.curves.find_one({"ainvs": minq_ainvs})["lmfdb_label"]
# We do not just do the following, as Sage's installed database
# might not have all the curves in the LMFDB database.
# minq_label = E.minimal_quadratic_twist()[0].label()
if "gens" in data:
generator = parse_gens(data["gens"])
if len(G) == 0:
tor_struct = "\mathrm{Trivial}"
tor_group = "\mathrm{Trivial}"
else:
tor_group = " \\times ".join(["\Z/{%s}\Z" % a.order() for a in G])
if "torsion_structure" in data:
info["tor_structure"] = " \\times ".join(["\Z/{%s}\Z" % int(a) for a in data["torsion_structure"]])
else:
info["tor_structure"] = tor_group
info.update(data)
if rank >= 2:
lder_tex = "L%s(E,1)" % ("^{(" + str(rank) + ")}")
elif rank == 1:
lder_tex = "L%s(E,1)" % ("'" * rank)
else:
assert rank == 0
lder_tex = "L(E,1)"
info["Gamma0optimal"] = cremona_label[-1] == "1" if cremona_iso_class != "990h" else cremona_label[-1] == "3"
info["modular_degree"] = modular_degree
p_adic_data_exists = (C.elliptic_curves.padic_db.find({"lmfdb_iso": lmfdb_iso_class}).count()) > 0 and info[
"Gamma0optimal"
]
# Local data
local_data = []
for p in N.prime_factors():
local_info = E.local_data(p, algorithm="generic")
local_data.append(
{
"p": p,
"tamagawa_number": local_info.tamagawa_number(),
"kodaira_symbol": web_latex(local_info.kodaira_symbol()).replace("$", ""),
"reduction_type": local_info.bad_reduction_type(),
}
)
mod_form_iso = lmfdb_label_regex.match(lmfdb_iso_class).groups()[1]
tamagawa_numbers = [E.local_data(p, algorithm="generic").tamagawa_number() for p in N.prime_factors()]
# if we use E.tamagawa_numbers() it calls E.local_data(p) which
# crashes on some curves e.g. 164411a1
info.update(
{
"conductor": N,
"disc_factor": latex(discriminant.factor()),
"j_invar_factor": j_inv_factored,
"label": lmfdb_label,
"cremona_label": cremona_label,
"iso_class": lmfdb_iso_class,
"cremona_iso_class": cremona_iso_class,
"equation": web_latex(E),
#'f': ajax_more(E.q_eigenform, 10, 20, 50, 100, 250),
"f": web_latex(E.q_eigenform(10)),
"generators": ", ".join(web_latex(g) for g in generator) if "gens" in data else " ",
"lder": lder_tex,
"p_adic_primes": [p for p in sage.all.prime_range(5, 100) if E.is_ordinary(p) and not p.divides(N)],
"p_adic_data_exists": p_adic_data_exists,
"ainvs": format_ainvs(data["ainvs"]),
"CM": CM,
"CMD": CMD,
"EndE": EndE,
"tamagawa_numbers": r" \cdot ".join(str(sage.all.factor(c)) for c in tamagawa_numbers),
"local_data": local_data,
"cond_factor": latex(N.factor()),
"xintegral_points": ", ".join(web_latex(P) for P in xintpoints),
"tor_gens": ", ".join(web_latex(eval(g)) for g in data["torsion_generators"])
if False
else ", ".join(web_latex(P.element().xy()) for P in list(G)),
}
)
info["friends"] = [
("Isogeny class " + lmfdb_iso_class, "/EllipticCurve/Q/%s" % lmfdb_iso_class),
("Minimal quadratic twist " + minq_label, "/EllipticCurve/Q/%s" % minq_label),
("All twists ", url_for("rational_elliptic_curves", jinv=jinv)),
("L-function", url_for("l_functions.l_function_ec_page", label=lmfdb_label)),
(
"Symmetric square L-function",
url_for("l_functions.l_function_ec_sym_page", power="2", label=lmfdb_iso_class),
),
(
"Symmetric 4th power L-function",
url_for("l_functions.l_function_ec_sym_page", power="4", label=lmfdb_iso_class),
),
]
info["friends"].append(
(
"Modular form " + lmfdb_iso_class.replace(".", ".2"),
url_for("emf.render_elliptic_modular_forms", level=int(N), weight=2, character=0, label=mod_form_iso),
)
)
info["downloads"] = [
("Download coeffients of q-expansion", url_for("download_EC_qexp", label=lmfdb_label, limit=100)),
("Download all stored data", url_for("download_EC_all", label=lmfdb_label)),
]
# info['learnmore'] = [('Elliptic Curves', url_for("not_yet_implemented"))]
# info['plot'] = image_src(plot)
info["plot"] = url_for("plot_ec", label=lmfdb_label)
properties2 = [
("Label", "%s" % lmfdb_label),
(None, '<img src="%s" width="200" height="150"/>' % url_for("plot_ec", label=lmfdb_label)),
("Conductor", "\(%s\)" % N),
("Discriminant", "\(%s\)" % discriminant),
("j-invariant", "%s" % web_latex(j_invariant)),
("CM", "%s" % CM),
("Rank", "\(%s\)" % rank),
("Torsion Structure", "\(%s\)" % tor_group),
]
# properties.extend([ "prop %s = %s<br/>" % (_,_*1923) for _ in range(12) ])
credit = "John Cremona"
if info["label"] == info["cremona_label"]:
t = "Elliptic Curve %s" % info["label"]
else:
t = "Elliptic Curve %s (Cremona label %s)" % (info["label"], info["cremona_label"])
bread = [("Elliptic Curves ", url_for("rational_elliptic_curves")), ("Elliptic curves %s" % lmfdb_label, " ")]
return render_template(
"elliptic_curve/elliptic_curve.html",
properties2=properties2,
credit=credit,
bread=bread,
title=t,
info=info,
friends=info["friends"],
downloads=info["downloads"],
)