def by_weierstrass(eqn):
w = weierstrass_eqn_regex.match(eqn)
if not w:
w = short_weierstrass_eqn_regex.match(eqn)
if not w:
return elliptic_curve_jump_error(eqn, {})
try:
ainvs = [ZZ(ai) for ai in w.groups()]
except TypeError:
return elliptic_curve_jump_error(eqn, {})
E = EllipticCurve(ainvs).global_minimal_model()
N = E.conductor()
ainvs = [str(ai) for ai in E.ainvs()]
data = db_ec().find_one({'ainvs': ainvs})
if data is None:
return elliptic_curve_jump_error(eqn, {'conductor':N}, missing_curve=True)
return redirect(url_for(".by_ec_label", label=data['lmfdb_label']), 301)
def by_weierstrass(eqn):
w = weierstrass_eqn_regex.match(eqn)
if not w:
w = short_weierstrass_eqn_regex.match(eqn)
if not w:
return elliptic_curve_jump_error(eqn, {})
try:
ainvs = [ZZ(ai) for ai in w.groups()]
except TypeError:
return elliptic_curve_jump_error(eqn, {})
E = EllipticCurve(ainvs).global_minimal_model()
N = E.conductor()
ainvs = [str(ai) for ai in E.ainvs()]
data = db_ec().find_one({'ainvs': ainvs})
if data is None:
return elliptic_curve_jump_error(eqn, {'conductor':N}, missing_curve=True)
return redirect(url_for(".by_ec_label", label=data['lmfdb_label']), 301)
def curve(self, P=None, verbose=False):
# Given a point on X_E(7), return the symplectically isomorphic curve.
if P==None:
return self.Eab
tP = tuple(P)
try:
return self.P2C[tP]
except KeyError:
pass
if verbose:
print("mapping P={} to its curve".format(P))
if self._symplectic and P==[0,1,0]:
E2 = self.Eab
if self._base==QQ:
E2 = E2.minimal_model()
self.P2C[tP] = [E2]
if verbose:
print("P={} is the base point, returning {}".format(P,E2.ainvs()))
return [E2]
ds = (d for d in (self.d11(P), self.d22(P), self.d33(P), self.d44(P)))
d = 0
while d==0:
d = ds.next()
if verbose: print("d={}".format(d))
if not self._symplectic:
d *= self.DeltaE()
c4 = self.C(P) * d**2
c6 = -self.K(P) * d**3
E2 = EllipticCurve([0,0,0,-c4/48,-c6/864])
assert E2.j_invariant()==self.j(P)
if self._base==QQ:
E2 = E2.minimal_model()
self.P2C[tP] = [E2]
if verbose:
print("P={} is a general point, returning {}".format(P,E2.ainvs()))
return [E2]
def check_curves(field_label='2.0.4.1',
min_norm=0,
max_norm=None,
label=None,
check_ap=False,
verbose=False):
r"""Go through all Bianchi Modular Forms with the given field label,
assumed imaginary quadratic (i.e. '2.0.d.1' with d in
{4,8,3,7,11}), check whether an elliptic curve exists with the
same label. If so, and if check_ap is True, check that the a_P agree.
"""
if field_label not in fields:
print("No BMF data available for field {}".format(field_label))
return
else:
K = field_from_label(field_label)
print("Checking forms over {}, norms from {} to {}".format(
field_label, min_norm, max_norm))
query = {}
query['field_label'] = field_label
query['dimension'] = 1 # only look at rational newforms
if label:
print("looking for {} only".format(label))
query['short_label'] = label # e.g. '91.1-a'
else:
query['level_norm'] = {'$gte': int(min_norm)}
if max_norm:
query['level_norm']['$lte'] = int(max_norm)
cursor = forms.search(query, sort=['level_norm'])
labels = [f['short_label'] for f in cursor]
nforms = len(labels)
print("found {} newforms".format(nforms))
labels = [lab for lab in labels if lab not in false_curves[field_label]]
nforms = len(labels)
print(
" of which {} should have associated curves (not false ones)".format(
nforms))
nfound = 0
nnotfound = 0
nok = 0
missing_curves = []
mismatches = []
primes = list(primes_iter(K, maxnorm=1000)) if check_ap else []
curve_ap = {} # curve_ap[conductor_label] will be a dict iso -> ap
form_ap = {} # form_ap[conductor_label] will be a dict iso -> ap
# Now look at all newforms, check that there is an elliptic
# curve of the same label, and if so compare ap-lists. The
# dicts curve_ap and form_ap store these when there is
# disagreement: e.g. curve_ap[conductor_label][iso_label] =
# aplist.
print("checking {} newforms".format(nforms))
n = 0
for curve_label in labels:
n += 1
if n % 100 == 0:
perc = 100.0 * n / nforms
print("{} forms checked ({}%)".format(n, perc))
# We find the forms again since otherwise the cursor might timeout during the loop.
label = "-".join([field_label, curve_label])
if verbose:
print("newform and isogeny class label {}".format(label))
f = forms.lucky({'label': label})
if f:
if verbose:
print("found newform with label {}".format(label))
else:
print("no newform in database has label {}!".format(label))
continue
ec = nfcurves.lucky({'class_label': label, 'number': 1})
if ec:
if verbose:
print("curve with label %s found in the database" %
curve_label)
nfound += 1
if not check_ap:
continue
ainvsK = parse_ainvs(K, ec['ainvs'])
if verbose:
print("E = {}".format(ainvsK))
E = EllipticCurve(ainvsK)
if verbose:
print("constructed elliptic curve {}".format(E.ainvs()))
good_flags = [E.has_good_reduction(P) for P in primes]
good_primes = [P for (P, flag) in zip(primes, good_flags) if flag]
if verbose:
print("{} good primes".format(len(good_primes)))
f_aplist = f['hecke_eigs']
f_aplist = [ap for ap, flag in zip(f_aplist, good_flags) if flag]
nap = len(f_aplist)
if verbose:
print("recovered {} ap from BMF".format(nap))
aplist = [
E.reduction(P).trace_of_frobenius() for P in good_primes[:nap]
]
if verbose:
print("computed {} ap from elliptic curve".format(nap))
if aplist[:nap] == f_aplist[:nap]:
nok += 1
if verbose:
print("Curve {} and newform agree! (checked {} ap)".format(
ec['short_label'], nap))
else:
print("Curve {} does NOT agree with newform".format(
ec['short_label']))
mismatches.append(label)
if verbose:
for P, aPf, aPc in zip(good_primes[:nap], f_aplist[:nap],
aplist[:nap]):
if aPf != aPc:
print("P = {} with norm {}".format(
P,
P.norm().factor()))
print("ap from curve: %s" % aPc)
print("ap from form: %s" % aPf)
if not ec['conductor_label'] in curve_ap:
curve_ap[ec['conductor_label']] = {}
form_ap[ec['conductor_label']] = {}
curve_ap[ec['conductor_label']][ec['iso_label']] = aplist
form_ap[ec['conductor_label']][f['label_suffix']] = f_aplist
else:
if verbose:
print("No curve with label %s found in the database!" %
curve_label)
missing_curves.append(f['short_label'])
nnotfound += 1
# Report progress:
n = nfound + nnotfound
if nnotfound:
print(
"Out of %s newforms, %s curves were found in the database and %s were not found"
% (n, nfound, nnotfound))
else:
print(
"Out of %s newforms, all %s had curves with the same label and ap"
% (n, nfound))
if nfound == nok:
print("All curves agree with matching newforms")
else:
print("%s curves agree with matching newforms, %s do not" %
(nok, nfound - nok))
if nnotfound:
print("%s missing curves" % len(missing_curves))
else:
pass
if mismatches:
print("{} form-curve pairs had inconsistent ap:".format(
len(mismatches)))
print(mismatches)
def find_curves(field_label='2.2.5.1', min_norm=0, max_norm=None, label=None, outfilename=None, verbose=False, effort=500):
r""" Go through all Hilbert Modular Forms with the given field label,
assumed totally real, for level norms in the given range, test
whether an elliptic curve exists with the same label; if not, find
the curves using Magma; output these to a file.
"""
print("Checking forms over {}, norms from {} to {}".format(field_label,min_norm,max_norm))
if outfilename:
print("Output of curves found to {}".format(outfilename))
else:
print("No curve search or output, just checking")
query = {}
query['field_label'] = field_label
if fields.find({'label': field_label}).count() == 0:
if verbose:
print("No HMF data for field %s" % field_label)
return None
query['dimension'] = 1 # only look at rational newforms
if label:
print("looking for {} only".format(label))
query['short_label'] = label # e.g. '91.1-a'
else:
query['level_norm'] = {'$gte': int(min_norm)}
if max_norm:
query['level_norm']['$lte'] = int(max_norm)
cursor = forms.find(query)
cursor.sort([('level_norm', pymongo.ASCENDING)])
labels = [f['label'] for f in cursor]
nfound = 0
nnotfound = 0
nok = 0
missing_curves = []
K = HilbertNumberField(field_label)
primes = [P['ideal'] for P in K.primes_iter(1000)]
curve_ap = {} # curve_ap[conductor_label] will be a dict iso -> ap
form_ap = {} # form_ap[conductor_label] will be a dict iso -> ap
# Step 1: look at all newforms, check that there is an elliptic
# curve of the same label, and if so compare ap-lists. The
# dicts curve_ap and form_ap store these when there is
# disagreement: e.g. curve_ap[conductor_label][iso_label] =
# aplist.
for curve_label in labels:
# We find the forms again since otherwise the cursor might timeout during the loop.
f = forms.find_one({'label': curve_label})
ec = nfcurves.find_one({'field_label': field_label, 'class_label': curve_label, 'number': 1})
if ec:
if verbose:
print("curve with label %s found in the database" % curve_label)
nfound += 1
ainvsK = parse_ainvs(K.K(), ec['ainvs'])
E = EllipticCurve(ainvsK)
good_flags = [E.has_good_reduction(P) for P in primes]
good_primes = [P for (P, flag) in zip(primes, good_flags) if flag]
aplist = [E.reduction(P).trace_of_frobenius() for P in good_primes]
f_aplist = [int(a) for a in f['hecke_eigenvalues']]
f_aplist = [ap for ap, flag in zip(f_aplist, good_flags) if flag]
nap = min(len(aplist), len(f_aplist))
if aplist[:nap] == f_aplist[:nap]:
nok += 1
if verbose:
print("Curve {} and newform agree! (checked {} ap)".format(ec['short_label'],nap))
else:
print("Curve {} does NOT agree with newform".format(ec['short_label']))
if verbose:
for P,aPf,aPc in zip(good_primes[:nap], f_aplist[:nap], aplist[:nap]):
if aPf!=aPc:
print("P = {} with norm {}".format(P,P.norm().factor()))
print("ap from curve: %s" % aPc)
print("ap from form: %s" % aPf)
if not ec['conductor_label'] in curve_ap:
curve_ap[ec['conductor_label']] = {}
form_ap[ec['conductor_label']] = {}
curve_ap[ec['conductor_label']][ec['iso_label']] = aplist
form_ap[ec['conductor_label']][f['label_suffix']] = f_aplist
else:
if verbose:
print("No curve with label %s found in the database!" % curve_label)
missing_curves.append(f['short_label'])
nnotfound += 1
# Report progress:
n = nfound + nnotfound
if nnotfound:
print("Out of %s newforms, %s curves were found in the database and %s were not found" % (n, nfound, nnotfound))
else:
print("Out of %s newforms, all %s had curves with the same label and ap" % (n, nfound))
if nfound == nok:
print("All curves agree with matching newforms")
else:
print("%s curves agree with matching newforms, %s do not" % (nok, nfound - nok))
if nnotfound:
print("%s missing curves" % len(missing_curves))
else:
return
# Step 2: for each newform for which there was no curve, call interface to Magma's EllipticCurveSearch()
# (unless outfilename is None in which case just dump the missing labels to a file)
if outfilename:
outfile = file(outfilename, mode="w")
else:
t = file("curves_missing.{}".format(field_label), mode="w")
for c in missing_curves:
t.write(c)
t.write("\n")
t.close()
return
def output(L):
if outfilename:
outfile.write(L)
if verbose:
sys.stdout.write(L)
bad_p = []
#if field_label=='4.4.1600.1': bad_p = [7**2,13**2,29**2]
if field_label=='4.4.2304.1': bad_p = [19**2,29**2]
if field_label=='4.4.4225.1': bad_p = [17**2,23**2]
if field_label=='4.4.7056.1': bad_p = [29**2,31**2]
if field_label=='4.4.7168.1': bad_p = [29**2]
if field_label=='4.4.9248.1': bad_p = [23**2]
if field_label=='4.4.11025.1': bad_p = [17**2,37**2,43**2]
if field_label=='4.4.13824.1': bad_p = [19**2]
if field_label=='4.4.12400.1': bad_p = [23**2]
if field_label=='4.4.180769.1': bad_p = [23**2]
if field_label=='6.6.905177.1': bad_p = [2**3]
bad_p = []
effort0 = effort
for nf_label in missing_curves:
if verbose:
print("Curve %s is missing from the database..." % nf_label)
form = forms.find_one({'field_label': field_label, 'short_label': nf_label})
if not form:
print("... form %s not found!" % nf_label)
else:
if verbose:
print("... found form, calling Magma search")
print("Conductor = %s" % form['level_ideal'].replace(" ",""))
N = K.ideal(form['level_label'])
neigs = len(form['hecke_eigenvalues'])
Plist = [P['ideal'] for P in K.primes_iter(neigs)]
goodP = [(i, P) for i, P in enumerate(Plist)
if not P.divides(N)
and not P.norm() in bad_p
and P.residue_class_degree()==1]
aplist = [int(form['hecke_eigenvalues'][i]) for i, P in goodP]
Plist = [P for i,P in goodP]
nap = len(Plist)
neigs0 = min(nap,100)
effort=effort0
if verbose:
print("Using %s ap from Hilbert newform and effort %s" % (neigs0,effort))
if bad_p:
print("( excluding primes with norms {})".format(bad_p))
#inds = list(set([randint(0,nap-1) for _ in range(neigs0)]))
inds = range(neigs0)
Plist0 = [Plist[i] for i in inds]
aplist0 = [aplist[i] for i in inds]
curves = EllipticCurveSearch(K.K(), Plist0, N, aplist0, effort)
# rep = 0
allrep=0
while not curves and allrep<10:
allrep += 1
effort*=2
# if rep<2:
# rep += 1
# else:
# rep = 1
# effort *=2
if verbose:
print("No curves found by Magma, trying again with effort %s..." % effort)
curves = EllipticCurveSearch(K.K(), Plist0, N, aplist0, effort)
if verbose:
if curves:
print("Success!")
else:
print("Still no success")
E = None
if curves:
E = curves[0]
print("%s curves for %s found, first is %s" % (len(curves),nf_label,E.ainvs()))
else:
print("!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!")
print("!!! No curves for %s found (using %s ap) !!!" % (nf_label,len(aplist)))
print("!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!")
if E!=None:
ec = {}
ec['field_label'] = field_label
ec['conductor_label'] = form['level_label']
ec['iso_label'] = form['label_suffix']
ec['number'] = int(1)
ec['conductor_ideal'] = form['level_ideal'].replace(" ","")
ec['conductor_norm'] = form['level_norm']
ai = E.ainvs()
ec['ainvs'] = ";".join([",".join([str(c) for c in list(a)]) for a in ai])
#print ec['ainvs']
ec['cm'] = '?'
ec['base_change'] = []
output(make_curves_line(ec) + "\n")
if outfilename:
outfile.flush()
def test_isom_labels(lab1,lab2, symplectic=True, verbose=False):
E1 = EllipticCurve(lab1)
E2 = EllipticCurve(lab2)
if verbose:
print("Testing {} ={} and {} = {}".format(lab1,E1.ainvs(),lab2,E2.ainvs()))
return test_isom(E1, E2, symplectic=symplectic, verbose=verbose)
def find_curves(field_label='2.2.5.1',
min_norm=0,
max_norm=None,
label=None,
outfilename=None,
verbose=False,
effort=500):
r""" Go through all Hilbert Modular Forms with the given field label,
assumed totally real, for level norms in the given range, test
whether an elliptic curve exists with the same label; if not, find
the curves using Magma; output these to a file.
"""
print("Checking forms over {}, norms from {} to {}".format(
field_label, min_norm, max_norm))
if outfilename:
print("Output of curves found to {}".format(outfilename))
else:
print("No curve search or output, just checking")
query = {}
query['field_label'] = field_label
if fields.find({'label': field_label}).count() == 0:
if verbose:
print("No HMF data for field %s" % field_label)
return None
query['dimension'] = 1 # only look at rational newforms
if label:
print("looking for {} only".format(label))
query['short_label'] = label # e.g. '91.1-a'
else:
query['level_norm'] = {'$gte': int(min_norm)}
if max_norm:
query['level_norm']['$lte'] = int(max_norm)
cursor = forms.find(query)
cursor.sort([('level_norm', pymongo.ASCENDING)])
labels = [f['label'] for f in cursor]
nfound = 0
nnotfound = 0
nok = 0
missing_curves = []
K = HilbertNumberField(field_label)
primes = [P['ideal'] for P in K.primes_iter(1000)]
curve_ap = {} # curve_ap[conductor_label] will be a dict iso -> ap
form_ap = {} # form_ap[conductor_label] will be a dict iso -> ap
# Step 1: look at all newforms, check that there is an elliptic
# curve of the same label, and if so compare ap-lists. The
# dicts curve_ap and form_ap store these when there is
# disagreement: e.g. curve_ap[conductor_label][iso_label] =
# aplist.
for curve_label in labels:
# We find the forms again since otherwise the cursor might timeout during the loop.
f = forms.find_one({'label': curve_label})
ec = nfcurves.find_one({
'field_label': field_label,
'class_label': curve_label,
'number': 1
})
if ec:
if verbose:
print("curve with label %s found in the database" %
curve_label)
nfound += 1
ainvsK = parse_ainvs(K.K(), ec['ainvs'])
E = EllipticCurve(ainvsK)
good_flags = [E.has_good_reduction(P) for P in primes]
good_primes = [P for (P, flag) in zip(primes, good_flags) if flag]
aplist = [E.reduction(P).trace_of_frobenius() for P in good_primes]
f_aplist = [int(a) for a in f['hecke_eigenvalues']]
f_aplist = [ap for ap, flag in zip(f_aplist, good_flags) if flag]
nap = min(len(aplist), len(f_aplist))
if aplist[:nap] == f_aplist[:nap]:
nok += 1
if verbose:
print("Curve {} and newform agree! (checked {} ap)".format(
ec['short_label'], nap))
else:
print("Curve {} does NOT agree with newform".format(
ec['short_label']))
if verbose:
for P, aPf, aPc in zip(good_primes[:nap], f_aplist[:nap],
aplist[:nap]):
if aPf != aPc:
print("P = {} with norm {}".format(
P,
P.norm().factor()))
print("ap from curve: %s" % aPc)
print("ap from form: %s" % aPf)
if not ec['conductor_label'] in curve_ap:
curve_ap[ec['conductor_label']] = {}
form_ap[ec['conductor_label']] = {}
curve_ap[ec['conductor_label']][ec['iso_label']] = aplist
form_ap[ec['conductor_label']][f['label_suffix']] = f_aplist
else:
if verbose:
print("No curve with label %s found in the database!" %
curve_label)
missing_curves.append(f['short_label'])
nnotfound += 1
# Report progress:
n = nfound + nnotfound
if nnotfound:
print(
"Out of %s newforms, %s curves were found in the database and %s were not found"
% (n, nfound, nnotfound))
else:
print(
"Out of %s newforms, all %s had curves with the same label and ap"
% (n, nfound))
if nfound == nok:
print("All curves agree with matching newforms")
else:
print("%s curves agree with matching newforms, %s do not" %
(nok, nfound - nok))
if nnotfound:
print("%s missing curves" % len(missing_curves))
else:
return
# Step 2: for each newform for which there was no curve, call interface to Magma's EllipticCurveSearch()
# (unless outfilename is None in which case just dump the missing labels to a file)
if outfilename:
outfile = file(outfilename, mode="w")
else:
t = file("curves_missing.{}".format(field_label), mode="w")
for c in missing_curves:
t.write(c)
t.write("\n")
t.close()
return
def output(L):
if outfilename:
outfile.write(L)
if verbose:
sys.stdout.write(L)
bad_p = []
#if field_label=='4.4.1600.1': bad_p = [7**2,13**2,29**2]
if field_label == '4.4.2304.1': bad_p = [19**2, 29**2]
if field_label == '4.4.4225.1': bad_p = [17**2, 23**2]
if field_label == '4.4.7056.1': bad_p = [29**2, 31**2]
if field_label == '4.4.7168.1': bad_p = [29**2]
if field_label == '4.4.9248.1': bad_p = [23**2]
if field_label == '4.4.11025.1': bad_p = [17**2, 37**2, 43**2]
if field_label == '4.4.13824.1': bad_p = [19**2]
if field_label == '4.4.12400.1': bad_p = [23**2]
if field_label == '4.4.180769.1': bad_p = [23**2]
if field_label == '6.6.905177.1': bad_p = [2**3]
bad_p = []
effort0 = effort
for nf_label in missing_curves:
if verbose:
print("Curve %s is missing from the database..." % nf_label)
form = forms.find_one({
'field_label': field_label,
'short_label': nf_label
})
if not form:
print("... form %s not found!" % nf_label)
else:
if verbose:
print("... found form, calling Magma search")
print("Conductor = %s" % form['level_ideal'].replace(" ", ""))
N = K.ideal(form['level_label'])
neigs = len(form['hecke_eigenvalues'])
Plist = [P['ideal'] for P in K.primes_iter(neigs)]
goodP = [(i, P) for i, P in enumerate(Plist)
if not P.divides(N) and not P.norm() in bad_p
and P.residue_class_degree() == 1]
aplist = [int(form['hecke_eigenvalues'][i]) for i, P in goodP]
Plist = [P for i, P in goodP]
nap = len(Plist)
neigs0 = min(nap, 100)
effort = effort0
if verbose:
print("Using %s ap from Hilbert newform and effort %s" %
(neigs0, effort))
if bad_p:
print("( excluding primes with norms {})".format(bad_p))
#inds = list(set([randint(0,nap-1) for _ in range(neigs0)]))
inds = range(neigs0)
Plist0 = [Plist[i] for i in inds]
aplist0 = [aplist[i] for i in inds]
curves = EllipticCurveSearch(K.K(), Plist0, N, aplist0, effort)
# rep = 0
allrep = 0
while not curves and allrep < 10:
allrep += 1
effort *= 2
# if rep<2:
# rep += 1
# else:
# rep = 1
# effort *=2
if verbose:
print(
"No curves found by Magma, trying again with effort %s..."
% effort)
curves = EllipticCurveSearch(K.K(), Plist0, N, aplist0, effort)
if verbose:
if curves:
print("Success!")
else:
print("Still no success")
E = None
if curves:
E = curves[0]
print("%s curves for %s found, first is %s" %
(len(curves), nf_label, E.ainvs()))
else:
print("!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!")
print("!!! No curves for %s found (using %s ap) !!!" %
(nf_label, len(aplist)))
print("!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!")
if E != None:
ec = {}
ec['field_label'] = field_label
ec['conductor_label'] = form['level_label']
ec['iso_label'] = form['label_suffix']
ec['number'] = int(1)
ec['conductor_ideal'] = form['level_ideal'].replace(" ", "")
ec['conductor_norm'] = form['level_norm']
ai = E.ainvs()
ec['ainvs'] = ";".join(
[",".join([str(c) for c in list(a)]) for a in ai])
#print ec['ainvs']
ec['cm'] = '?'
ec['base_change'] = []
output(make_curves_line(ec) + "\n")
if outfilename:
outfile.flush()
def find_curves(field_label='2.2.5.1', min_norm=0, max_norm=None, outfilename=None, verbose=False):
r""" Go through all Hilbert Modular Forms with the given field label,
assumed totally real, for level norms in the given range, test
whether an elliptic curve exists with the same label; if not, find
the curves using Magma; output these to a file.
"""
query = {}
query['field_label'] = field_label
if fields.find({'label': field_label}).count() == 0:
if verbose:
print("No HMF data for field %s" % field_label)
return None
query['dimension'] = 1 # only look at rational newforms
query['level_norm'] = {'$gte': int(min_norm)}
if max_norm:
query['level_norm']['$lte'] = int(max_norm)
else:
max_norm = 'infinity'
cursor = forms.find(query)
cursor.sort([('level_norm', pymongo.ASCENDING)])
labels = [f['label'] for f in cursor]
nfound = 0
nnotfound = 0
nok = 0
missing_curves = []
K = HilbertNumberField(field_label)
primes = [P['ideal'] for P in K.primes_iter(100)]
curve_ap = {} # curve_ap[conductor_label] will be a dict iso -> ap
form_ap = {} # form_ap[conductor_label] will be a dict iso -> ap
# Step 1: look at all newforms, check that there is an elliptic
# curve of the same label, and if so compare ap-lists. The
# dicts curve_ap and form_ap store these when there is
# disagreement: e.g. curve_ap[conductor_label][iso_label] =
# aplist.
for curve_label in labels:
# We find the forms again since otherwise the cursor might timeout during the loop.
f = forms.find_one({'label': curve_label})
ec = nfcurves.find_one({'field_label': field_label, 'class_label': curve_label, 'number': 1})
if ec:
if verbose:
print("curve with label %s found in the database" % curve_label)
nfound += 1
ainvsK = [K.K()([QQ(str(c)) for c in ai]) for ai in ec['ainvs']]
E = EllipticCurve(ainvsK)
good_flags = [E.has_good_reduction(P) for P in primes]
good_primes = [P for (P, flag) in zip(primes, good_flags) if flag]
aplist = [E.reduction(P).trace_of_frobenius() for P in good_primes[:30]]
f_aplist = [int(a) for a in f['hecke_eigenvalues'][:40]]
f_aplist = [ap for ap, flag in zip(f_aplist, good_flags) if flag][:30]
if aplist == f_aplist:
nok += 1
if verbose:
print("Curve %s and newform agree!" % ec['short_label'])
else:
print("Curve %s does NOT agree with newform" % ec['short_label'])
if verbose:
print("ap from curve: %s" % aplist)
print("ap from form: %s" % f_aplist)
if not ec['conductor_label'] in curve_ap:
curve_ap[ec['conductor_label']] = {}
form_ap[ec['conductor_label']] = {}
curve_ap[ec['conductor_label']][ec['iso_label']] = aplist
form_ap[ec['conductor_label']][f['label_suffix']] = f_aplist
else:
if verbose:
print("No curve with label %s found in the database!" % curve_label)
missing_curves.append(f['short_label'])
nnotfound += 1
# Report progress:
n = nfound + nnotfound
if nnotfound:
print("Out of %s newforms, %s curves were found in the database and %s were not found" % (n, nfound, nnotfound))
else:
print("Out of %s newforms, all %s had curves with the same label and ap" % (n, nfound))
if nfound == nok:
print("All curves agree with matching newforms")
else:
print("%s curves agree with matching newforms, %s do not" % (nok, nfound - nok))
if nnotfound:
print("%s missing curves" % len(missing_curves))
else:
return
# Step 2: for each newform for which there was no curve, call interface to Magma's EllipticCurveSearch()
if outfilename:
outfile = file(outfilename, mode="w")
def output(L):
if outfilename:
outfile.write(L)
if verbose:
sys.stdout.write(L)
for nf_label in missing_curves:
if verbose:
print("Curve %s is missing from the database..." % nf_label)
form = forms.find_one({'field_label': field_label, 'short_label': nf_label})
if not form:
print("... form %s not found!" % nf_label)
else:
if verbose:
print("... found form, calling Magma search")
N = K.ideal(form['level_label'])
neigs = len(form['hecke_eigenvalues'])
if verbose:
print("Using %s ap from Hilbert newform" % neigs)
Plist = [P['ideal'] for P in K.primes_iter(neigs)]
goodP = [(i, P) for i, P in enumerate(Plist) if not P.divides(N)]
aplist = [int(form['hecke_eigenvalues'][i]) for i, P in goodP]
curves = EllipticCurveSearch(K.K(), Plist, N, aplist)
if not curves:
if verbose:
print("No curves found by Magma, trying again...")
curves = EllipticCurveSearch(K.K(), Plist, N, aplist)
if verbose:
if curves:
print("Success!")
else:
print("Still no success, giving up")
#curves = EllipticCurveSearch(K.K(), [], N, [])
E = None
if curves:
E = curves[0]
print("%s curves for %s found, first is %s" % (len(curves),nf_label,E.ainvs()))
else:
print("!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!")
print("!!! No curves for %s found (using %s ap) !!!" % (nf_label,len(aplist)))
print("!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!")
if E!=None:
ec = {}
ec['field_label'] = field_label
ec['conductor_label'] = form['level_label']
ec['iso_label'] = form['label_suffix']
ec['number'] = int(1)
ec['conductor_ideal'] = form['level_ideal'].replace(" ","")
ec['conductor_norm'] = form['level_norm']
ai = E.ainvs()
ec['ainvs'] = [[str(c) for c in list(a)] for a in ai]
ec['cm'] = '?'
ec['base_change'] = []
output(make_curves_line(ec) + "\n")
if outfilename:
outfile.flush()
def find_curves(field_label='2.2.5.1',
min_norm=0,
max_norm=None,
outfilename=None,
verbose=False):
r""" Go through all Hilbert Modular Forms with the given field label,
assumed totally real, for level norms in the given range, test
whether an elliptic curve exists with the same label; if not, find
the curves using Magma; output these to a file.
"""
query = {}
query['field_label'] = field_label
if fields.find({'label': field_label}).count() == 0:
if verbose:
print("No HMF data for field %s" % field_label)
return None
query['dimension'] = 1 # only look at rational newforms
query['level_norm'] = {'$gte': int(min_norm)}
if max_norm:
query['level_norm']['$lte'] = int(max_norm)
else:
max_norm = 'infinity'
cursor = forms.find(query)
cursor.sort([('level_norm', pymongo.ASCENDING)])
labels = [f['label'] for f in cursor]
nfound = 0
nnotfound = 0
nok = 0
missing_curves = []
K = HilbertNumberField(field_label)
primes = [P['ideal'] for P in K.primes_iter(100)]
curve_ap = {} # curve_ap[conductor_label] will be a dict iso -> ap
form_ap = {} # form_ap[conductor_label] will be a dict iso -> ap
# Step 1: look at all newforms, check that there is an elliptic
# curve of the same label, and if so compare ap-lists. The
# dicts curve_ap and form_ap store these when there is
# disagreement: e.g. curve_ap[conductor_label][iso_label] =
# aplist.
for curve_label in labels:
# We find the forms again since otherwise the cursor might timeout during the loop.
f = forms.find_one({'label': curve_label})
ec = nfcurves.find_one({
'field_label': field_label,
'class_label': curve_label,
'number': 1
})
if ec:
if verbose:
print("curve with label %s found in the database" %
curve_label)
nfound += 1
ainvsK = [K.K()([QQ(str(c)) for c in ai]) for ai in ec['ainvs']]
E = EllipticCurve(ainvsK)
good_flags = [E.has_good_reduction(P) for P in primes]
good_primes = [P for (P, flag) in zip(primes, good_flags) if flag]
aplist = [
E.reduction(P).trace_of_frobenius() for P in good_primes[:30]
]
f_aplist = [int(a) for a in f['hecke_eigenvalues'][:40]]
f_aplist = [ap for ap, flag in zip(f_aplist, good_flags)
if flag][:30]
if aplist == f_aplist:
nok += 1
if verbose:
print("Curve %s and newform agree!" % ec['short_label'])
else:
print("Curve %s does NOT agree with newform" %
ec['short_label'])
if verbose:
print("ap from curve: %s" % aplist)
print("ap from form: %s" % f_aplist)
if not ec['conductor_label'] in curve_ap:
curve_ap[ec['conductor_label']] = {}
form_ap[ec['conductor_label']] = {}
curve_ap[ec['conductor_label']][ec['iso_label']] = aplist
form_ap[ec['conductor_label']][f['label_suffix']] = f_aplist
else:
if verbose:
print("No curve with label %s found in the database!" %
curve_label)
missing_curves.append(f['short_label'])
nnotfound += 1
# Report progress:
n = nfound + nnotfound
if nnotfound:
print(
"Out of %s newforms, %s curves were found in the database and %s were not found"
% (n, nfound, nnotfound))
else:
print(
"Out of %s newforms, all %s had curves with the same label and ap"
% (n, nfound))
if nfound == nok:
print("All curves agree with matching newforms")
else:
print("%s curves agree with matching newforms, %s do not" %
(nok, nfound - nok))
if nnotfound:
print("%s missing curves" % len(missing_curves))
else:
return
# Step 2: for each newform for which there was no curve, call interface to Magma's EllipticCurveSearch()
if outfilename:
outfile = file(outfilename, mode="w")
def output(L):
if outfilename:
outfile.write(L)
if verbose:
sys.stdout.write(L)
for nf_label in missing_curves:
if verbose:
print("Curve %s is missing from the database..." % nf_label)
form = forms.find_one({
'field_label': field_label,
'short_label': nf_label
})
if not form:
print("... form %s not found!" % nf_label)
else:
if verbose:
print("... found form, calling Magma search")
N = K.ideal(form['level_label'])
neigs = len(form['hecke_eigenvalues'])
if verbose:
print("Using %s ap from Hilbert newform" % neigs)
Plist = [P['ideal'] for P in K.primes_iter(neigs)]
goodP = [(i, P) for i, P in enumerate(Plist) if not P.divides(N)]
aplist = [int(form['hecke_eigenvalues'][i]) for i, P in goodP]
curves = EllipticCurveSearch(K.K(), Plist, N, aplist)
if not curves:
if verbose:
print("No curves found by Magma, trying again...")
curves = EllipticCurveSearch(K.K(), Plist, N, aplist)
if verbose:
if curves:
print("Success!")
else:
print("Still no success, giving up")
#curves = EllipticCurveSearch(K.K(), [], N, [])
E = None
if curves:
E = curves[0]
print("%s curves for %s found, first is %s" %
(len(curves), nf_label, E.ainvs()))
else:
print("!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!")
print("!!! No curves for %s found (using %s ap) !!!" %
(nf_label, len(aplist)))
print("!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!")
if E != None:
ec = {}
ec['field_label'] = field_label
ec['conductor_label'] = form['level_label']
ec['iso_label'] = form['label_suffix']
ec['number'] = int(1)
ec['conductor_ideal'] = form['level_ideal'].replace(" ", "")
ec['conductor_norm'] = form['level_norm']
ai = E.ainvs()
ec['ainvs'] = [[str(c) for c in list(a)] for a in ai]
ec['cm'] = '?'
ec['base_change'] = []
output(make_curves_line(ec) + "\n")
if outfilename:
outfile.flush()
