def add_isogeny_matrices(N1, N2):
"""
Add the 'isogeny_matrix' field to every curve in the database
whose conductor is between N1 and N2 inclusive. The matrix is
stored as a list of n lists of n ints, where n is the size of the
class and the (i,j) entry is the degree of a cyclic isogeny from
curve i to curve j in the class, using the lmfdb numbering within
each class. Hence this matrix is exactly the same for all curves
in the class. This was added in July 2014 to save recomputing the
complete isogeny class every time despite the fact that the curves
in the class were already in the database.
"""
query = {}
query['conductor'] = {'$gt': int(N1) - 1, '$lt': int(N2) + 1}
query['number'] = int(1)
res = curves.find(query)
res = res.sort([('conductor', pymongo.ASCENDING),
('lmfdb_iso', pymongo.ASCENDING)])
for C in res:
if 'isogeny_matrix' in C:
print("class {} has isogeny matrix already, skipping".format(
C['label']))
continue
lmfdb_iso = C['lmfdb_iso']
E = EllipticCurve([int(a) for a in C['ainvs']])
M = E.isogeny_class(order="lmfdb").matrix()
mat = [list([int(c) for c in r]) for r in M.rows()]
n = len(mat)
print("{} curves in class {}".format(n, lmfdb_iso))
for label_i in [lmfdb_iso + str(i + 1) for i in range(n)]:
data = {}
data['lmfdb_label'] = label_i
data['isogeny_matrix'] = mat
curves.update({'lmfdb_label': label_i}, {"$set": data},
upsert=True)
def add_isogeny_matrices(N1,N2):
"""
Add the 'isogeny_matrix' field to every curve in the database
whose conductor is between N1 and N2 inclusive. The matrix is
stored as a list of n lists of n ints, where n is the size of the
class and the (i,j) entry is the degree of a cyclic isogeny from
curve i to curve j in the class, using the lmfdb numbering within
each class. Hence this matrix is exactly the same for all curves
in the class. This was added in July 2014 to save recomputing the
complete isogeny class every time despite the fact that the curves
in the class were already in the database.
"""
query = {}
query['conductor'] = { '$gt': int(N1)-1, '$lt': int(N2)+1 }
query['number'] = int(1)
res = curves.find(query)
res = res.sort([('conductor', pymongo.ASCENDING),
('lmfdb_iso', pymongo.ASCENDING)])
for C in res:
if 'isogeny_matrix' in C:
print("class {} has isogeny matrix already, skipping".format(C['label']))
continue
lmfdb_iso = C['lmfdb_iso']
E = EllipticCurve([int(a) for a in C['ainvs']])
M = E.isogeny_class(order="lmfdb").matrix()
mat = [list([int(c) for c in r]) for r in M.rows()]
n = len(mat)
print("{} curves in class {}".format(n,lmfdb_iso))
for label_i in [lmfdb_iso+str(i+1) for i in range(n)]:
data = {}
data['lmfdb_label'] = label_i
data['isogeny_matrix'] = mat
curves.update({'lmfdb_label': label_i}, {"$set": data}, upsert=True)
def render_isogeny_class(iso_class):
info = {}
credit = 'John Cremona'
lmfdb_iso = iso_class # e.g. '11.a'
N, iso, number = lmfdb_label_regex.match(lmfdb_iso).groups()
CDB = lmfdb.base.getDBConnection().elliptic_curves.curves
E1data = CDB.find_one({'lmfdb_label': lmfdb_iso + '1'})
if E1data is None:
return elliptic_curve_jump_error(lmfdb_iso, {})
cremona_iso = E1data['iso']
ainvs = [int(a) for a in E1data['ainvs']]
E1 = EllipticCurve(ainvs)
ver = sage.version.version.split('.') # e.g. "6.1.beta2"
ma = int(ver[0])
mi = int(ver[1])
if ma > 6 or ma == 6 and mi > 1:
# Code for Sage 6.2 and later:
isogeny_class = E1.isogeny_class()
curves = isogeny_class.curves
mat = isogeny_class.matrix()
else:
# Code for Sage 6.1 and before:
curves, mat = E1.isogeny_class()
size = len(curves)
# Create a list of the curves in the class from the database, so
# they are in the correct order!
db_curves = [E1]
optimal_flags = [False] * size
degrees = [0] * size
if 'degree' in E1data:
degrees[0] = E1data['degree']
else:
try:
degrees[0] = E1.modular_degree()
except RuntimeError:
pass
cremona_labels = [E1data['label']] + [0] * (size - 1)
if E1data['number'] == 1:
optimal_flags[0] = True
for i in range(2, size + 1):
Edata = CDB.find_one({'lmfdb_label': lmfdb_iso + str(i)})
E = EllipticCurve([int(a) for a in Edata['ainvs']])
cremona_labels[i - 1] = Edata['label']
if Edata['number'] == 1:
optimal_flags[i - 1] = True
if 'degree' in Edata:
degrees[i - 1] = Edata['degree']
else:
try:
degrees[i - 1] = E.modular_degree()
except RuntimeError:
pass
db_curves.append(E)
if cremona_iso == '990h': # this isogeny class is labeled wrong in Cremona's tables
optimal_flags = [False, False, True, False]
# Now work out the permutation needed to match the two lists of curves:
perm = [db_curves.index(E) for E in curves]
# Apply the same permutation to the isogeny matrix:
mat = [[mat[perm[i], perm[j]] for j in range(size)] for i in range(size)]
info = {'label': lmfdb_iso}
info['optimal_ainvs'] = ainvs
info['rank'] = E1data['rank']
info['isogeny_matrix'] = latex(matrix(mat))
# info['f'] = ajax_more(E.q_eigenform, 10, 20, 50, 100, 250)
info['f'] = web_latex(E.q_eigenform(10))
info['graph_img'] = url_for('.plot_iso_graph', label=lmfdb_iso)
info['curves'] = [[
lmfdb_iso + str(i + 1), cremona_labels[i],
str(list(c.ainvs())),
c.torsion_order(), degrees[i], optimal_flags[i]
] for i, c in enumerate(db_curves)]
friends = []
# friends.append(('Quadratic Twist', "/quadratic_twists/%s" % (lmfdb_iso)))
friends.append(('L-function',
url_for("l_functions.l_function_ec_page",
label=lmfdb_iso)))
friends.append(('Symmetric square L-function',
url_for("l_functions.l_function_ec_sym_page",
power='2',
label=lmfdb_iso)))
friends.append(('Symmetric 4th power L-function',
url_for("l_functions.l_function_ec_sym_page",
power='4',
label=lmfdb_iso)))
# render_one_elliptic_modular_form(level,weight,character,label,**kwds)
friends.append(('Modular form ' + lmfdb_iso.replace('.', '.2'),
url_for("emf.render_elliptic_modular_forms",
level=N,
weight=2,
character=0,
label=iso)))
info['friends'] = friends
info['downloads'] = [('Download coeffients of q-expansion',
url_for(".download_EC_qexp",
label=lmfdb_iso,
limit=100)),
('Download stored data for curves in this class',
url_for(".download_EC_all", label=lmfdb_iso))]
if lmfdb_iso == cremona_iso:
t = "Elliptic Curve Isogeny Class %s" % lmfdb_iso
else:
t = "Elliptic Curve Isogeny Class %s (Cremona label %s)" % (
lmfdb_iso, cremona_iso)
bread = [('Elliptic Curves ', url_for(".rational_elliptic_curves")),
('isogeny class %s' % lmfdb_iso, ' ')]
return render_template("iso_class.html",
info=info,
bread=bread,
credit=credit,
title=t,
friends=info['friends'],
downloads=info['downloads'])
class ECisog_class(object):
"""
Class for an isogeny class of elliptic curves 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)
self.make_class()
@staticmethod
def by_label(label):
"""
Searches for a specific elliptic curve isogeny class in the
curves collection by its label, which can be either a curve
label (e.g. "11.a1") or a class label (e.g. "11.a") in either
LMFDB or Cremona format.
"""
#print "label = %s" % label
try:
N, iso, number = split_lmfdb_label(label)
if number:
data = db_ec().find_one({"lmfdb_label": label})
else:
data = db_ec().find_one({"lmfdb_label": label + "1"})
except AttributeError:
try:
N, iso, number = split_cremona_label(label)
if number:
data = db_ec().find_one({"label": label})
else:
data = db_ec().find_one({"label": label + "1"})
except AttributeError:
return "Invalid label" # caller must catch this and raise an error
if data:
return ECisog_class(data)
return "Class not found" # caller must catch this and raise an error
def make_class(self):
self.ainvs_str = self.ainvs
self.ainvs = [int(a) for a in self.ainvs_str]
self.E = EllipticCurve(self.ainvs)
self.CM = self.E.has_cm()
try:
# Extract the isogeny degree matrix from the database
size = len(self.isogeny_matrix)
from sage.matrix.all import Matrix
self.isogeny_matrix = Matrix(self.isogeny_matrix)
except AttributeError:
# Failsafe: construct it from scratch
self.isogeny_matrix = self.E.isogeny_class(order="lmfdb").matrix()
size = self.isogeny_matrix.nrows()
self.ncurves = size
# Create isogeny graph:
self.graph = make_graph(self.isogeny_matrix)
P = self.graph.plot(edge_labels=True)
self.graph_img = encode_plot(P)
self.graph_link = '<img src="%s" width="200" height="150"/>' % self.graph_img
# Create a list of the curves in the class from the database
self.db_curves = [self.E]
self.optimal_flags = [False] * size
self.degrees = [0] * size
if self.degree:
self.degrees[0] = self.degree
else:
try:
self.degrees[0] = self.E.modular_degree()
except RuntimeError:
pass
# Fill in the curves in the class by looking each one up in the db:
self.cremona_labels = [self.label] + [0] * (size - 1)
if self.number == 1:
self.optimal_flags[0] = True
for i in range(2, size + 1):
Edata = db_ec().find_one({'lmfdb_label': self.lmfdb_iso + str(i)})
Ei = EllipticCurve([int(a) for a in Edata['ainvs']])
self.cremona_labels[i - 1] = Edata['label']
if Edata['number'] == 1:
self.optimal_flags[i - 1] = True
if 'degree' in Edata:
self.degrees[i - 1] = Edata['degree']
else:
try:
self.degrees[i - 1] = Ei.modular_degree()
except RuntimeError:
pass
self.db_curves.append(Ei)
if self.iso == '990h': # this isogeny class is labeled wrong in Cremona's tables
self.optimal_flags = [False, False, True, False]
self.isogeny_matrix_str = latex(matrix(self.isogeny_matrix))
N, iso, number = split_lmfdb_label(self.lmfdb_iso)
self.newform = web_latex(self.E.q_eigenform(10))
self.newform_label = newform_label(N, 2, 1, iso)
self.newform_link = url_for("emf.render_elliptic_modular_forms",
level=N,
weight=2,
character=1,
label=iso)
newform_exists_in_db = is_newform_in_db(self.newform_label)
self.lfunction_link = url_for("l_functions.l_function_ec_page",
label=self.lmfdb_iso)
self.curves = [
dict([('label', self.lmfdb_iso + str(i + 1)),
('url',
url_for(".by_triple_label",
conductor=N,
iso_label=iso,
number=i + 1)),
('cremona_label', self.cremona_labels[i]),
('ainvs', str(list(c.ainvs()))),
('torsion', c.torsion_order()), ('degree', self.degrees[i]),
('optimal', self.optimal_flags[i])])
for i, c in enumerate(self.db_curves)
]
self.friends = [('L-function', self.lfunction_link)]
if not self.CM:
if int(N) <= 300:
self.friends += [('Symmetric square L-function',
url_for("l_functions.l_function_ec_sym_page",
power='2',
label=self.lmfdb_iso))]
if int(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.properties = [('Label', self.lmfdb_iso),
('Number of curves', str(self.ncurves)),
('Conductor', '\(%s\)' % N), ('CM', '%s' % self.CM),
('Rank', '\(%s\)' % self.rank), ('Graph', ''),
(None, self.graph_link)]
self.downloads = [('Download coefficients of newform',
url_for(".download_EC_qexp",
label=self.lmfdb_iso,
limit=100)),
('Download stored data for all curves',
url_for(".download_EC_all", label=self.lmfdb_iso))]
if self.lmfdb_iso == self.iso:
self.title = "Elliptic Curve Isogeny Class %s" % self.lmfdb_iso
else:
self.title = "Elliptic Curve Isogeny Class %s (Cremona label %s)" % (
self.lmfdb_iso, self.iso)
self.bread = [('Elliptic Curves', url_for("ecnf.index")),
('$\Q$', url_for(".rational_elliptic_curves")),
('%s' % N, url_for(".by_conductor", conductor=N)),
('%s' % iso, ' ')]
class ECisog_class(object):
"""
Class for an isogeny class of elliptic curves 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)
self.make_class()
@staticmethod
def by_label(label):
"""
Searches for a specific elliptic curve isogeny class in the
curves collection by its label, which can be either a curve
label (e.g. "11.a1") or a class label (e.g. "11.a") in either
LMFDB or Cremona format.
"""
#print "label = %s" % label
try:
N, iso, number = split_lmfdb_label(label)
if number:
data = db_ec().find_one({"lmfdb_label" : label})
else:
data = db_ec().find_one({"lmfdb_label" : label+"1"})
except AttributeError:
try:
N, iso, number = split_cremona_label(label)
if number:
data = db_ec().find_one({"label" : label})
else:
data = db_ec().find_one({"label" : label+"1"})
except AttributeError:
return "Invalid label" # caller must catch this and raise an error
if data:
return ECisog_class(data)
return "Class not found" # caller must catch this and raise an error
def make_class(self):
self.ainvs_str = self.ainvs
self.ainvs = [int(a) for a in self.ainvs_str]
self.E = EllipticCurve(self.ainvs)
self.CM = self.E.has_cm()
try:
# Extract the isogeny degree matrix from the database
size = len(self.isogeny_matrix)
from sage.matrix.all import Matrix
self.isogeny_matrix = Matrix(self.isogeny_matrix)
except AttributeError:
# Failsafe: construct it from scratch
self.isogeny_matrix = self.E.isogeny_class(order="lmfdb").matrix()
size = self.isogeny_matrix.nrows()
self.ncurves = size
# Create isogeny graph:
self.graph = make_graph(self.isogeny_matrix)
P = self.graph.plot(edge_labels=True)
self.graph_img = encode_plot(P)
self.graph_link = '<img src="%s" width="200" height="150"/>' % self.graph_img
# Create a list of the curves in the class from the database
self.db_curves = [self.E]
self.optimal_flags = [False] * size
self.degrees = [0] * size
if self.degree:
self.degrees[0] = self.degree
else:
try:
self.degrees[0] = self.E.modular_degree()
except RuntimeError:
pass
# Fill in the curves in the class by looking each one up in the db:
self.cremona_labels = [self.label] + [0] * (size - 1)
if self.number == 1:
self.optimal_flags[0] = True
for i in range(2, size + 1):
Edata = db_ec().find_one({'lmfdb_label': self.lmfdb_iso + str(i)})
Ei = EllipticCurve([int(a) for a in Edata['ainvs']])
self.cremona_labels[i - 1] = Edata['label']
if Edata['number'] == 1:
self.optimal_flags[i - 1] = True
if 'degree' in Edata:
self.degrees[i - 1] = Edata['degree']
else:
try:
self.degrees[i - 1] = Ei.modular_degree()
except RuntimeError:
pass
self.db_curves.append(Ei)
if self.iso == '990h': # this isogeny class is labeled wrong in Cremona's tables
self.optimal_flags = [False, False, True, False]
self.isogeny_matrix_str = latex(matrix(self.isogeny_matrix))
N, iso, number = split_lmfdb_label(self.lmfdb_iso)
self.newform = web_latex(self.E.q_eigenform(10))
self.newform_label = newform_label(N,2,1,iso)
self.newform_link = url_for("emf.render_elliptic_modular_forms", level=N, weight=2, character=1, label=iso)
self.newform_exists_in_db = is_newform_in_db(self.newform_label)
self.lfunction_link = url_for("l_functions.l_function_ec_page", label=self.lmfdb_iso)
self.curves = [dict([('label',self.lmfdb_iso + str(i + 1)),
('url',url_for(".by_triple_label", conductor=N, iso_label=iso, number=i+1)),
('cremona_label',self.cremona_labels[i]),
('ainvs',str(list(c.ainvs()))),
('torsion',c.torsion_order()),
('degree',self.degrees[i]),
('optimal',self.optimal_flags[i])])
for i, c in enumerate(self.db_curves)]
self.friends = [('L-function', self.lfunction_link)]
if not self.CM:
if int(N)<=300:
self.friends += [('Symmetric square L-function', url_for("l_functions.l_function_ec_sym_page", power='2', label=self.lmfdb_iso))]
if int(N)<=50:
self.friends += [('Symmetric cube L-function', url_for("l_functions.l_function_ec_sym_page", power='3', label=self.lmfdb_iso))]
if self.newform_exists_in_db:
self.friends += [('Modular form ' + self.newform_label, self.newform_link)]
self.properties = [('Label', self.lmfdb_iso),
('Number of curves', str(self.ncurves)),
('Conductor', '\(%s\)' % N),
('CM', '%s' % self.CM),
('Rank', '\(%s\)' % self.rank),
('Graph', ''),(None, self.graph_link)
]
self.downloads = [('Download coefficients of newform', url_for(".download_EC_qexp", label=self.lmfdb_iso, limit=1000)),
('Download stored data for all curves', url_for(".download_EC_all", label=self.lmfdb_iso))]
if self.lmfdb_iso == self.iso:
self.title = "Elliptic Curve Isogeny Class %s" % self.lmfdb_iso
else:
self.title = "Elliptic Curve Isogeny Class %s (Cremona label %s)" % (self.lmfdb_iso, self.iso)
self.bread = [('Elliptic Curves', url_for("ecnf.index")),
('$\Q$', url_for(".rational_elliptic_curves")),
('%s' % N, url_for(".by_conductor", conductor=N)),
('%s' % iso, ' ')]
self.code = {}
self.code['show'] = {'sage':''} # use default show names
self.code['class'] = {'sage':'E = EllipticCurve("%s1")\n'%(self.lmfdb_iso) + 'E.isogeny_class()\n'}
self.code['curves'] = {'sage':'E.isogeny_class().curves'}
self.code['rank'] = {'sage':'E.rank()'}
self.code['q_eigenform'] = {'sage':'E.q_eigenform(10)'}
self.code['matrix'] = {'sage':'E.isogeny_class().matrix()'}
self.code['plot'] = {'sage':'E.isogeny_graph().plot(edge_labels=True)'}
def render_isogeny_class(iso_class):
info = {}
credit = 'John Cremona'
lmfdb_iso = iso_class # e.g. '11.a'
N, iso, number = lmfdb_label_regex.match(lmfdb_iso).groups()
CDB = lmfdb.base.getDBConnection().elliptic_curves.curves
E1data = CDB.find_one({'lmfdb_label': lmfdb_iso + '1'})
if E1data is None:
return elliptic_curve_jump_error(lmfdb_iso, {})
cremona_iso = E1data['iso']
ainvs = [int(a) for a in E1data['ainvs']]
E1 = EllipticCurve(ainvs)
curves, mat = E1.isogeny_class()
size = len(curves)
# Create a list of the curves in the class from the database, so
# they are in the correct order!
db_curves = [E1]
optimal_flags = [False] * size
degrees = [0] * size
if 'degree' in E1data:
degrees[0] = E1data['degree']
else:
try:
degrees[0] = E1.modular_degree()
except RuntimeError:
pass
cremona_labels = [E1data['label']] + [0] * (size - 1)
if E1data['number'] == 1:
optimal_flags[0] = True
for i in range(2, size + 1):
Edata = CDB.find_one({'lmfdb_label': lmfdb_iso + str(i)})
E = EllipticCurve([int(a) for a in Edata['ainvs']])
cremona_labels[i - 1] = Edata['label']
if Edata['number'] == 1:
optimal_flags[i - 1] = True
if 'degree' in Edata:
degrees[i - 1] = Edata['degree']
else:
try:
degrees[i - 1] = E.modular_degree()
except RuntimeError:
pass
db_curves.append(E)
if cremona_iso == '990h': # this isogeny class is labeled wrong in Cremona's tables
optimal_flags = [False, False, True, False]
# Now work out the permutation needed to match the two lists of curves:
perm = [db_curves.index(E) for E in curves]
# Apply the same permutation to the isogeny matrix:
mat = [[mat[perm[i], perm[j]] for j in range(size)] for i in range(size)]
info = {'label': lmfdb_iso}
info['optimal_ainvs'] = ainvs
info['rank'] = E1data['rank']
info['isogeny_matrix'] = latex(matrix(mat))
# info['f'] = ajax_more(E.q_eigenform, 10, 20, 50, 100, 250)
info['f'] = web_latex(E.q_eigenform(10))
info['graph_img'] = url_for('plot_iso_graph', label=lmfdb_iso)
info['curves'] = [[lmfdb_iso + str(i + 1), cremona_labels[i], str(
list(c.ainvs())), c.torsion_order(), degrees[i], optimal_flags[i]] for i, c in enumerate(db_curves)]
friends = []
# friends.append(('Quadratic Twist', "/quadratic_twists/%s" % (lmfdb_iso)))
friends.append(('L-function', url_for("l_functions.l_function_ec_page", label=lmfdb_iso)))
friends.append(('Symmetric square L-function', url_for("l_functions.l_function_ec_sym_page",
power='2', label=lmfdb_iso)))
friends.append(('Symmetric 4th power L-function', url_for("l_functions.l_function_ec_sym_page",
power='4', label=lmfdb_iso)))
# render_one_elliptic_modular_form(level,weight,character,label,**kwds)
friends.append(('Modular form ' + lmfdb_iso.replace(
'.', '.2'), url_for("emf.render_elliptic_modular_forms", level=N, weight=2, character=0, label=iso)))
info['friends'] = friends
info['downloads'] = [('Download coeffients of q-expansion', url_for("download_EC_qexp", label=lmfdb_iso, limit=100)),
('Download stored data for curves in this class', url_for("download_EC_all", label=lmfdb_iso))]
if lmfdb_iso == cremona_iso:
t = "Elliptic Curve Isogeny Class %s" % lmfdb_iso
else:
t = "Elliptic Curve Isogeny Class %s (Cremona label %s)" % (lmfdb_iso, cremona_iso)
bread = [('Elliptic Curves ', url_for("rational_elliptic_curves")), ('isogeny class %s' % lmfdb_iso, ' ')]
return render_template("elliptic_curve/iso_class.html", info=info, bread=bread, credit=credit, title=t, friends=info['friends'], downloads=info['downloads'])
def class_js(lab):
E = EllipticCurve(lab)
return Set([E1.j_invariant() for E1 in E.isogeny_class()])
