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 = 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 = 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"],
)
