def solve_ideal_equation(gamma, I, D, N, O):
"""Find mu_0 in Rj such that (O* gamma / NO)[mu_0] = I / NO.
Args:
gamma: An element of O.
I: A left O-ideal.
D: The index [O : R + Rj].
N: The norm of I. Must be prime.
O: An order in a rational quaternion algebra containing 1, i, j, k.
Returns:
mu_0 in Rj such that 0 != gamma * mu_0 in I.
"""
assert N == I.norm()
assert is_prime(N)
# d = D*c + N*_
d, c, _ = xgcd(D, N)
assert d == 1
a, b = [Integer(x) for x in O.quaternion_algebra().invariants()]
F = GF(N)
# The suffix _ff means that this element is over a finite field, not the
# rationals.
B_ff = QuaternionAlgebra(F, a, b)
i_ff, j_ff, k_ff = B_ff.gens()
# phi is homomorphism O -> (a, b | F) with kernel NO.
def phi(alpha):
# alpha + NO = alpha_prime + NO.
alpha_prime = D * c * alpha
t, x, y, z = [
Integer(coeff) for coeff in alpha_prime.coefficient_tuple()
]
return t + x * i_ff + y * j_ff + z * k_ff
gamma_ff = phi(gamma)
gamma_ff_mat = gamma_ff.matrix(action="left")
I_basis_ff = [phi(alpha).coefficient_tuple() for alpha in I.basis()]
# Only use 2nd and 3rd rows of gamma_ff_mat so solution is in Rj.
lin_system = matrix(F, [gamma_ff_mat[2], gamma_ff_mat[3]] + I_basis_ff)
sol = lin_system.left_kernel().basis()[0]
y, z = sol[0], sol[1]
mu_ff = y * j_ff + z * k_ff
assert vector(F, (gamma_ff * mu_ff).coefficient_tuple()) in span(
I_basis_ff, F)
B = O.quaternion_algebra()
i, j, k = B.gens()
mu_0 = sum(
Integer(coeff) * elem
for coeff, elem in zip(mu_ff.coefficient_tuple(), [1, i, j, k]))
assert 0 != mu_0
assert gamma * mu_0 in I
return mu_0
def complete_column(a, c, i=1):
"""
Completes a column a,c with gcd(a,c) = 1 to an SL_2(Z)-matrix.
INPUT:
- a - int, the top entry of the column.
- c - int, the bottom entry of the column.
- i - int.
OUTPUT:
- A matrix in SL_2(Z) with i-th column [a,c]
"""
if gcd(a, c) > 1:
raise ValueError('Input needs to be coprime')
_, d, b = xgcd(a, c)
b = -b
if i == 1:
return Matrix([[a, b], [c, d]])
elif i == 2:
return Matrix([[-b, a], [-d, c]])
def __add__(self, other):
if isinstance(other, Integer):
other *= IdentityEndomorphismMod(self.E, self.psi)
if isinstance(other, ZeroEndomorphismMod):
return self
if self.fx == other.fx:
if self.fy == other.fy:
m = (3 * self.fx ** 2 + self.A) / (2 * self.fy * self.f)
else:
return ZeroEndomorphismMod(self.E, self.psi)
else:
div, inv, _ = xgcd(self.fx.parent().lift(self.fx - other.fx), (self.psi))
if div.is_constant():
m = (self.fy - other.fy) * inv
else:
raise ZeroDivisionError(div)
fx = m ** 2 * self.f - self.fx - other.fx
fy = m * (self.fx - fx) - self.fy
return EndomorphismMod(self.E, fx, fy, self.psi)
def AL(N, S):
r"""
INPUT:
- N - a positive integer
- S - a maximal divisor of N or a set of primes
OUPUT:
- A matrix representing the partial Atkin-Lehner operator W_S
"""
try:
S = prod([p**(N.valuation(p)) for p in S])
except TypeError:
if gcd(S, N / S) > 1:
S = S.prime_divisors()
S = prod([p**(N.valuation(p)) for p in S])
_, w, z = xgcd(S, N / S)
z = -z
return Matrix([[S, 1], [N * z, S * w]])
def find_monic_replacements(self, p, t, pt_generators, prev_nu):
r"""
Replace possibly non-monic generators of `N_{(p^t)}(B)` by monic
generators.
INPUT:
- ``p`` -- a prime element of `D`
- ``t`` -- a non-negative integer
- ``pt_generators`` -- a list `(g_1, \ldots, g_s)` of polynomials in
`D[X]` such that `N_{(p^t)}(B) = (g_1, \ldots, g_s) + pN_{(p^{t-1})}(B)`
- ``prev_nu`` -- a `(p^{t-1})`-minimal polynomial of `B`
OUTPUT:
A list `(h_1, \ldots, h_r)` of monic polynomials such that
`N_{(p^t)}(B) = (h_1, \ldots, h_r) + pN_{(p^{t-1})}(B)`.
EXAMPLES::
sage: from sage.matrix.compute_J_ideal import ComputeMinimalPolynomials
sage: B = matrix(ZZ, [[1, 0, 1], [1, -2, -1], [10, 0, 0]])
sage: C = ComputeMinimalPolynomials(B)
sage: x = polygen(ZZ, 'x')
sage: nu_1 = x^2 + x
sage: generators_4 = [2*x^2 + 2*x, x^2 + 3*x + 2]
sage: C.find_monic_replacements(2, 2, generators_4, nu_1)
[x^2 + 3*x + 2]
TESTS::
sage: C.find_monic_replacements(2, 3, generators_4, nu_1)
Traceback (most recent call last):
...
ValueError: [2*x^2 + 2*x, x^2 + 3*x + 2] not in N_{(2^3)}(B)
sage: C.find_monic_replacements(2, 2, generators_4, x^2)
Traceback (most recent call last):
...
ValueError: x^2 not in N_{(2^1)}(B)
ALGORITHM:
[HR2016]_, Algorithms 2 and 3.
"""
from sage.arith.misc import xgcd
if not all((g(self._B) % p**t).is_zero() for g in pt_generators):
raise ValueError("%s not in N_{(%s^%s)}(B)" %
(pt_generators, p, t))
if not (prev_nu(self._B) % p**(t - 1)).is_zero():
raise ValueError("%s not in N_{(%s^%s)}(B)" % (prev_nu, p, t - 1))
(X, ) = self._DX.gens()
replacements = []
for f in pt_generators:
g = f
p_prt = p_part(g, p)
while g != p * p_prt:
r = p_prt.quo_rem(prev_nu)[1]
g1 = g - p * p_prt
d, u, v = xgcd(g1.leading_coefficient(), p)
h = u * (p * r + g1) + v * p * prev_nu * X**(g1.degree() -
prev_nu.degree())
replacements.append(h % p**t)
#reduce coefficients mod p^t to keep coefficients small
g = g.quo_rem(h)[1]
p_prt = p_part(g, p)
replacements = list(set(replacements))
assert all(g.is_monic() for g in replacements),\
"Something went wrong in find_monic_replacements"
return replacements
def find_monic_replacements(self, p, t, pt_generators, prev_nu):
r"""
Replace possibly non-monic generators of `N_{(p^t)}(B)` by monic
generators.
INPUT:
- ``p`` -- a prime element of `D`
- ``t`` -- a non-negative integer
- ``pt_generators`` -- a list `(g_1, \ldots, g_s)` of polynomials in
`D[X]` such that `N_{(p^t)}(B) = (g_1, \ldots, g_s) + pN_{(p^{t-1})}(B)`
- ``prev_nu`` -- a `(p^{t-1})`-minimal polynomial of `B`
OUTPUT:
A list `(h_1, \ldots, h_r)` of monic polynomials such that
`N_{(p^t)}(B) = (h_1, \ldots, h_r) + pN_{(p^{t-1})}(B)`.
EXAMPLES::
sage: from sage.matrix.compute_J_ideal import ComputeMinimalPolynomials
sage: B = matrix(ZZ, [[1, 0, 1], [1, -2, -1], [10, 0, 0]])
sage: C = ComputeMinimalPolynomials(B)
sage: x = polygen(ZZ, 'x')
sage: nu_1 = x^2 + x
sage: generators_4 = [2*x^2 + 2*x, x^2 + 3*x + 2]
sage: C.find_monic_replacements(2, 2, generators_4, nu_1)
[x^2 + 3*x + 2]
TESTS::
sage: C.find_monic_replacements(2, 3, generators_4, nu_1)
Traceback (most recent call last):
...
ValueError: [2*x^2 + 2*x, x^2 + 3*x + 2] not in N_{(2^3)}(B)
sage: C.find_monic_replacements(2, 2, generators_4, x^2)
Traceback (most recent call last):
...
ValueError: x^2 not in N_{(2^1)}(B)
ALGORITHM:
[HR2016]_, Algorithms 2 and 3.
"""
from sage.arith.misc import xgcd
if not all((g(self._B) % p**t).is_zero()
for g in pt_generators):
raise ValueError("%s not in N_{(%s^%s)}(B)" %
(pt_generators, p, t))
if not (prev_nu(self._B) % p**(t-1)).is_zero():
raise ValueError("%s not in N_{(%s^%s)}(B)" % (prev_nu, p, t-1))
(X,) = self._DX.gens()
replacements = []
for f in pt_generators:
g = f
p_prt = p_part(g, p)
while g != p*p_prt:
r = p_prt.quo_rem(prev_nu)[1]
g1 = g - p*p_prt
d, u, v = xgcd(g1.leading_coefficient(), p)
h = u*(p*r + g1) + v*p*prev_nu*X**(g1.degree()-prev_nu.degree())
replacements.append(h % p**t)
#reduce coefficients mod p^t to keep coefficients small
g = g.quo_rem(h)[1]
p_prt = p_part(g, p)
replacements = list(set(replacements))
assert all(g.is_monic() for g in replacements),\
"Something went wrong in find_monic_replacements"
return replacements