def factorize_matrix(m,M):
#assert is_in_Gamma_1(m,M,determinant_condition = False)
assert m.det().abs() == 1
a,b,c,d = m.list()
if QQ(a).denominator() != 1:
raise RuntimeError
#return [m]
assert a % M == 1
aabs = ZZ(a).abs()
Zm = Zmod(M)
for alphamul in sorted(range(-aabs.sqrt(50)/M,aabs.sqrt(50)/M),key = lambda x: ZZ(x).abs()):
alpha = 1 + M*alphamul
if alpha.abs() >= aabs:
continue
delta0 = (Zm(alpha)**(-1)).lift()
for delta in xrange(delta0-10*M,delta0+10*M,M):
if alpha * delta == 1:
continue
gamma0 = ZZ( (alpha*delta -1) / M)
c1vec = gamma0.divisors()
for c1 in c1vec:
ulentry = (a*delta-b*c1*M).abs()
urentry = (b*alpha-a*gamma0/c1).abs()
if ulentry < aabs and urentry < b.abs():
gamma = c1*M
beta = ZZ(gamma0/c1)
r1 = Matrix(QQ,2,2,[alpha,beta,gamma,delta])
r2 = m*r1.adjugate()
assert r1.determinant() == 1
assert is_in_Gamma_1(r1,M,determinant_condition = False)
assert is_in_Gamma_1(r1,M,determinant_condition = False)
V1 = factorize_matrix(r1,M)
V2 = factorize_matrix(r2,M)
return V2 + V1
return [m]
def factorize_matrix(m,M):
#assert is_in_Gamma_1(m,M,determinant_condition = False)
assert m.det().abs() == 1
a,b,c,d = m.list()
if QQ(a).denominator() != 1:
raise RuntimeError
#return [m]
assert a % M == 1
aabs = ZZ(a).abs()
Zm = Zmod(M)
for alphamul in sorted(range(-aabs.sqrt(50)/M,aabs.sqrt(50)/M),key = lambda x: ZZ(x).abs()):
alpha = 1 + M*alphamul
if alpha.abs() >= aabs:
continue
delta0 = (Zm(alpha)**(-1)).lift()
for delta in xrange(delta0-10*M,delta0+10*M,M):
if alpha * delta == 1:
continue
gamma0 = ZZ( (alpha*delta -1) / M)
c1vec = gamma0.divisors()
for c1 in c1vec:
ulentry = (a*delta-b*c1*M).abs()
urentry = (b*alpha-a*gamma0/c1).abs()
if ulentry < aabs and urentry < b.abs():
gamma = c1*M
beta = ZZ(gamma0/c1)
r1 = Matrix(QQ,2,2,[alpha,beta,gamma,delta])
r2 = m*r1.adjoint()
assert r1.determinant() == 1
assert is_in_Gamma_1(r1,M,determinant_condition = False)
assert is_in_Gamma_1(r1,M,determinant_condition = False)
V1 = factorize_matrix(r1,M)
V2 = factorize_matrix(r2,M)
return V2 + V1
return [m]
def find_matrix_from_cusp(self, cusp):
r'''
Returns a matrix gamma and a cusp representative modulo Gamma0(N) (c2:d2),
represented as a matrix (a,b;c,d), such that gamma * cusp = (c2:d2).
'''
a, c = cusp
reduction_table, _ = self.cusp_reduction_table()
P = self.get_P1List()
if hasattr(P.N(),'number_field'):
K = P.N().number_field()
else:
K = QQ
# Find a matrix g = [a,b,c,d] in SL2(O_K) such that g * a/c = oo
# Define (c1:d1) to be the rep in P1(O_K/N) such that (c1:d1) == (c:d).
if c == 0: ## case cusp infinity: (a,c) should equal (1,0)
a = 1
g = Matrix(2,2,[1,0,0,1])
c1, d1 = P.normalize(0, 1)
else:
if K == QQ:
g0, d, b = ZZ(a).xgcd(-c)
if g0 != 1:
a /= g0
c /= g0
else:
"""
Compute gcd if a,c are coprime in F, and x,y such that
ax + cy = 1.
"""
if a.parent() != c.parent():
raise ValueError('a,c not in the same field.')
if a.gcd(c) != 1:
raise ValueError('a,c not coprime.')
d = next(o for o in K.ideal(c).residues() if a * o - 1 in K.ideal(c))
b = (a * d - 1) / c
g = Matrix(2,2,[[d,-b],[-c,a]]) # the inverse
c1, d1 = P.normalize(c, d)
assert g.determinant() == 1
A, T = reduction_table[(c1,d1)]
gamma = A.parent()(A * T * g)
return gamma, A
