def new_decomposition(self, verbose=False):
"""
Return complete irreducible Hecke decomposition of new subspace of self.
"""
V = self.degeneracy_matrix().kernel()
p = next_prime_of_characteristic_coprime_to(F.ideal(1), self.level())
T = self.hecke_matrix(p)
D = T.decomposition_of_subspace(V)
while len([X for X in D if not X[1]]) > 0:
p = next_prime_of_characteristic_coprime_to(p, self.level())
if verbose: print p.norm()
T = self.hecke_matrix(p)
D2 = []
for X in D:
if X[1]:
D2.append(X)
else:
if verbose: print T.restrict(X[0]).fcp()
for Z in T.decomposition_of_subspace(X[0]):
D2.append(Z)
D = D2
D = [self.subspace(X[0]) for X in D]
D.sort()
S = Sequence(D, immutable=True, cr=True, universe=int, check=False)
return S
def decomposition(self, B, verbose=False):
"""
Return Hecke decomposition of self using Hecke operators T_p
coprime to the level with norm(p) <= B.
"""
# TODO: rewrite to use primes_of_bounded_norm so that we
# iterate through primes ordered by *norm*, which is
# potentially vastly faster. Delete these functions
# involving characteristic!
p = next_prime_of_characteristic_coprime_to(F.ideal(1), self.level())
T = self.hecke_matrix(p)
D = T.decomposition()
while len([X for X in D if not X[1]]) > 0:
p = next_prime_of_characteristic_coprime_to(p, self.level())
if p.norm() > B:
break
if verbose: print p.norm()
T = self.hecke_matrix(p)
D2 = []
for X in D:
if X[1]:
D2.append(X)
else:
if verbose: print T.restrict(X[0]).fcp()
for Z in T.decomposition_of_subspace(X[0]):
D2.append(Z)
D = D2
D = [self.subspace(X[0]) for X in D]
D.sort()
S = Sequence(D, immutable=True, cr=True, universe=int, check=False)
return S
def new_decomposition(self, verbose=False):
"""
Return complete irreducible Hecke decomposition of new subspace of self.
"""
V = self.degeneracy_matrix().kernel()
p = next_prime_of_characteristic_coprime_to(F.ideal(1), self.level())
T = self.hecke_matrix(p)
D = T.decomposition_of_subspace(V)
while len([X for X in D if not X[1]]) > 0:
p = next_prime_of_characteristic_coprime_to(p, self.level())
if verbose: print p.norm()
T = self.hecke_matrix(p)
D2 = []
for X in D:
if X[1]:
D2.append(X)
else:
if verbose: print T.restrict(X[0]).fcp()
for Z in T.decomposition_of_subspace(X[0]):
D2.append(Z)
D = D2
D = [self.subspace(X[0]) for X in D]
D.sort()
S = Sequence(D, immutable=True, cr=True, universe=int, check=False)
return S
def decomposition(self, B, verbose=False):
"""
Return Hecke decomposition of self using Hecke operators T_p
coprime to the level with norm(p) <= B.
"""
# TODO: rewrite to use primes_of_bounded_norm so that we
# iterate through primes ordered by *norm*, which is
# potentially vastly faster. Delete these functions
# involving characteristic!
p = next_prime_of_characteristic_coprime_to(F.ideal(1), self.level())
T = self.hecke_matrix(p)
D = T.decomposition()
while len([X for X in D if not X[1]]) > 0:
p = next_prime_of_characteristic_coprime_to(p, self.level())
if p.norm() > B:
break
if verbose: print p.norm()
T = self.hecke_matrix(p)
D2 = []
for X in D:
if X[1]:
D2.append(X)
else:
if verbose: print T.restrict(X[0]).fcp()
for Z in T.decomposition_of_subspace(X[0]):
D2.append(Z)
D = D2
D = [self.subspace(X[0]) for X in D]
D.sort()
S = Sequence(D, immutable=True, cr=True, universe=int, check=False)
return S
def next_prime_not_dividing(P, I):
while True:
p = P.smallest_integer()
if p == 1:
Q = F.ideal(2)
elif p % 5 in [2,3]: # inert
Q = F.primes_above(next_prime(p))[0]
elif p == 5:
Q = F.ideal(7)
else: # p split
A = F.primes_above(p)
if A[0] == P:
Q = A[1]
else:
Q = F.primes_above(next_prime(p))[0]
if not Q.divides(I):
return Q
else:
P = Q # try again
def next_prime_not_dividing(P, I):
while True:
p = P.smallest_integer()
if p == 1:
Q = F.ideal(2)
elif p % 5 in [2,3]: # inert
Q = F.primes_above(next_prime(p))[0]
elif p == 5:
Q = F.ideal(7)
else: # p split
A = F.primes_above(p)
if A[0] == P:
Q = A[1]
else:
Q = F.primes_above(next_prime(p))[0]
if not Q.divides(I):
return Q
else:
P = Q # try again
def elliptic_curve_factors(self):
D = [X for X in self.new_decomposition() if X.dimension() == 1]
# Have to get rid of the Eisenstein factor
p = next_prime_of_characteristic_coprime_to(F.ideal(1), self.level())
while True:
q = p.residue_field().cardinality() + 1
E = [A for A in D if A.hecke_matrix(p)[0,0] == q]
if len(E) == 0:
break
elif len(E) == 1:
D = [A for A in D if A != E[0]]
break
else:
p = next_prime_of_characteristic_coprime_to(p, self.level())
return Sequence([EllipticCurveFactor(X, number) for number, X in enumerate(D)],
immutable=True, cr=True, universe=int, check=False)
def elliptic_curve_factors(self):
D = [X for X in self.new_decomposition() if X.dimension() == 1]
# Have to get rid of the Eisenstein factor
p = next_prime_of_characteristic_coprime_to(F.ideal(1), self.level())
while True:
q = p.residue_field().cardinality() + 1
E = [A for A in D if A.hecke_matrix(p)[0,0] == q]
if len(E) == 0:
break
elif len(E) == 1:
D = [A for A in D if A != E[0]]
break
else:
p = next_prime_of_characteristic_coprime_to(p, self.level())
return Sequence([EllipticCurveFactor(X, number) for number, X in enumerate(D)],
immutable=True, cr=True, universe=int, check=False)