def _div_mod_pO(self, other, base_multiply, p):
"""
ideal division modulo pO by using base_multiply
"""
n = other.row
m = other.column
F_p = finitefield.FinitePrimeField(p)
#Algo 2.3.7
if self == None:
self_new = matrix.zeroMatrix(n, F_p)
r = 0
else:
self_new = self
r = self.column
if r == m:
return matrix.unitMatrix(n, F_p)
# if r > m, raise NoInverseImage error
X = matrix.Subspace.fromMatrix(other.inverseImage(self_new))
B = X.supplementBasis()
other_self = other * B # first r columns are self, others are supplement
other_self.toFieldMatrix()
gamma_part = other_self.subMatrix(
range(1, n + 1), range(r + 1, m + 1))
omega_gamma = other_self.inverseImage(_matrix_mul_by_base_mul(
matrix.unitMatrix(n, F_p), gamma_part, base_multiply))
vect_list = []
for k in range(1, n + 1):
vect = vector.Vector([F_p.zero] * ((m - r) ** 2))
for i in range(1, m - r + 1):
for j in range(1, m - r + 1):
vect[(m - r) * (i - 1) + j] = _mul_place(
k, i, omega_gamma, m - r)[j + r]
vect_list.append(vect)
return matrix.FieldMatrix( (m - r) ** 2, n, vect_list).kernel()
def _div_mod_pO(self, other, base_multiply, p):
"""
ideal division modulo pO by using base_multiply
"""
n = other.row
m = other.column
F_p = finitefield.FinitePrimeField(p)
#Algo 2.3.7
if self == None:
self_new = matrix.zeroMatrix(n, F_p)
r = 0
else:
self_new = self
r = self.column
if r == m:
return matrix.unitMatrix(n, F_p)
# if r > m, raise NoInverseImage error
X = matrix.Subspace.fromMatrix(other.inverseImage(self_new))
B = X.supplementBasis()
other_self = other * B # first r columns are self, others are supplement
other_self.toFieldMatrix()
gamma_part = other_self.subMatrix(
range(1, n + 1), range(r + 1, m + 1))
omega_gamma = other_self.inverseImage(_matrix_mul_by_base_mul(
matrix.unitMatrix(n, F_p), gamma_part, base_multiply))
vect_list = []
for k in range(1, n + 1):
vect = vector.Vector([F_p.zero] * ((m - r) ** 2))
for i in range(1, m - r + 1):
for j in range(1, m - r + 1):
vect[(m - r) * (i - 1) + j] = _mul_place(
k, i, omega_gamma, m - r)[j + r]
vect_list.append(vect)
return matrix.FieldMatrix( (m - r) ** 2, n, vect_list).kernel()
def __init__(self, field, stconsts, name):
self.stconsts = stconsts
self.field = field
self.dim = stconsts[0].row
self.unit = matrix.unitMatrix(self.dim, self.field)
self.name = name
self.Rad = None
def algebra_example(dim, n):
# Algebras are defined over GF(2)
field = GF2
# All the algebras are unital
stconsts = [matrix.unitMatrix(dim, field)]
if dim == 1:
if n == 1:
# 1D, semisimple, 1 * 1 = 1
return AlgebraDescription(field, stconsts, True)
if dim == 2:
if n == 1:
# 2D, not semisimple, identity 1, aa = 1
stconsts.append(matrix.Matrix(2, 2, [(0,1),(1,0)], field))
return AlgebraDescription(field, stconsts, False)
if n == 2:
# 2D, not semisimple, identity 1, aa = 0
stconsts.append(matrix.Matrix(2, 2, [(0,1),(0,0)], field))
return AlgebraDescription(field, stconsts, False)
if dim == 3:
if n == 1:
# 3D, semisimple, identity 1, aa = a, bb = b, ab = ba = 0
stconsts.append(matrix.Matrix(3, 3, [(0,1,0),(0,1,0),(0,0,0)], field))
stconsts.append(matrix.Matrix(3, 3, [(0,0,1),(0,0,0),(0,0,1)], field))
return AlgebraDescription(field, stconsts, True)
raise ValueError("No such example")
def _check_H(p, H, field, gamma_mul):
"""
If H express the prime ideal, return (H, f),
where f: residual degree.
Else return some column of M_1 (not (1,0,..,0)^t).
"""
F_p = finitefield.FinitePrimeField(p)
if H == None:
f = field.degree
else:
f = H.row - H.column # rank(A), A.column
# CCANT Algo 6.2.9 step 10-11
# step 10
B = matrix.unitMatrix(f, F_p)
M_basis = []
for j in range(1, f + 1):
alpha_pow = _pow_by_base_mul(B[j], p, gamma_mul, f)
alpha_pow -= B[j]
M_basis.append(alpha_pow)
M_1 = matrix.FieldMatrix(f, f, M_basis).kernel()
# step 11
if M_1.column > 1: # dim(M_1) > 1
return M_1[M_1.column]
else:
H_simple = _two_element_repr_prime_ideal(
p, H.map(lambda x: x.getResidue()), field, f)
p_alg = algfield.BasicAlgNumber([[p] + [0] * (field.degree - 1), 1],
field.polynomial)
sol = module.Ideal_with_generator([p_alg, H_simple])
return (sol, f)
def _check_H(p, H, field, gamma_mul):
"""
If H express the prime ideal, return (H, f),
where f: residual degree.
Else return some column of M_1 (not (1,0,..,0)^t).
"""
F_p = finitefield.FinitePrimeField(p)
if H == None:
f = field.degree
else:
f = H.row - H.column # rank(A), A.column
# CCANT Algo 6.2.9 step 10-11
# step 10
B = matrix.unitMatrix(f, F_p)
M_basis = []
for j in range(1, f + 1):
alpha_pow = _pow_by_base_mul(B[j], p, gamma_mul, f)
alpha_pow -= B[j]
M_basis.append(alpha_pow)
M_1 = matrix.FieldMatrix(f, f, M_basis).kernel()
# step 11
if M_1.column > 1: # dim(M_1) > 1
return M_1[M_1.column]
else:
H_simple = _two_element_repr_prime_ideal(
p, H.map(lambda x: x.getResidue()), field, f)
p_alg = algfield.BasicAlgNumber([[p] + [0] * (field.degree - 1), 1],
field.polynomial)
sol = module.Ideal_with_generator([p_alg, H_simple])
return (sol, f)
def testMainCase(self):
# cubic_first [1, 9, 0, 3]
field = algfield.NumberField(self.cubic_first)
# p=3
cubic_first_3_index = [(1, 1), (2, 1)] # split & ramify
cubic_first_3_prime = 3 * module.Ideal(
matrix.unitMatrix(field.degree), field, field.integer_ring())
result_e_f, result_mul = self.result_transform(prime_decomp(3, self.cubic_first))
self.assertEqual(result_e_f, cubic_first_3_index)
self.assertEqual(result_mul, cubic_first_3_prime)
# cubic_second [1, 8, 0, 3]
field = algfield.NumberField(self.cubic_second)
# p=5
cubic_second_5_index = [(1, 1), (1, 2)] # split & inert
cubic_second_5_prime = 5 * module.Ideal(
matrix.unitMatrix(field.degree), field, field.integer_ring())
result_e_f, result_mul = self.result_transform(prime_decomp(5, self.cubic_second))
self.assertEqual(result_e_f, cubic_second_5_index)
self.assertEqual(result_mul, cubic_second_5_prime)
# sextic_first [7, 6, 5, 4, 3, 2, 1]
field = algfield.NumberField(self.sextic_first)
# p=2
sextic_first_2_index = [(2, 1), (4, 1)] # split & ramify
sextic_first_2_prime = 2 * module.Ideal(
matrix.unitMatrix(field.degree), field, field.integer_ring())
result_e_f, result_mul = self.result_transform(prime_decomp(2, self.sextic_first))
self.assertEqual(result_e_f, sextic_first_2_index)
self.assertEqual(result_mul, sextic_first_2_prime)
# sextic_second [6480, 1296, -252, 36, -7, 1]
field = algfield.NumberField(self.sextic_second)
# p=2
sextic_second_2_index = [(1, 1), (1, 4)] # split & ramify
sextic_second_2_prime = 2 * module.Ideal(
matrix.unitMatrix(field.degree), field, field.integer_ring())
result_e_f, result_mul = self.result_transform(prime_decomp(2, self.sextic_second))
self.assertEqual(result_e_f, sextic_second_2_index)
self.assertEqual(result_mul, sextic_second_2_prime)
# p=3
sextic_second_3_index = [(1, 1), (1, 1), (1, 1), (1, 2)] # split & inert
sextic_second_3_prime = 3 * module.Ideal(
matrix.unitMatrix(field.degree), field, field.integer_ring())
result_e_f, result_mul = self.result_transform(prime_decomp(3, self.sextic_second))
self.assertEqual(result_e_f, sextic_second_3_index)
self.assertEqual(result_mul, sextic_second_3_prime)
def __pow__(self, other):
if other < 0:
return Module.__pow__(self.inverse(), -other)
elif other == 0:
field = self.number_field
n = field.degree
int_basis = field.integer_ring()
return Ideal([matrix.unitMatrix(n), 1], field, int_basis)
else:
return Module.__pow__(self, other)
def __pow__(self, other):
if other < 0:
return Module.__pow__(self.inverse(), -other)
elif other == 0:
field = self.number_field
n = field.degree
int_basis = field.integer_ring()
return Ideal([matrix.unitMatrix(n), 1], field, int_basis)
else:
return Module.__pow__(self, other)
def _two_element_repr_prime_ideal(p, hnf, field, f):
"""
return alpha such that (p, alpha) is same ideal to
the ideal represented by hnf (matrix) w.r.t. the integer ring of field.
we assume the ideal is a prime ideal whose norm is p^f, where p is prime.
"""
# assume that first basis of the integral basis equals 1
int_basis = field.integer_ring()
# step 1
n = field.degree
Z = rational.theIntegerRing
beta = (p * matrix.unitMatrix(n, Z).subMatrix(
range(1, n + 1), range(2, n + 1))).copy()
beta.extendColumn(hnf)
m = beta.column
# step 2
R = 0
P_norm = p ** f
while True:
R += 1
lmd = [R] * m
while True:
# step 3
alpha = lmd[0] * beta[1]
for j in range(1, m):
alpha += lmd[j] * beta[j + 1]
alpha = _vector_to_algnumber(int_basis * alpha, field)
n = alpha.norm() / P_norm
r = divmod(int(n), p)[1]
if r:
return alpha
# step 4
for j in range(m - 1, 0, -1):
if lmd[j] != -R:
break
lmd[j] -= 1
for i in range(j + 1, m):
lmd[i] = R
# step 5
for j in range(m):
if lmd[j] != 0:
break
else:
break
def _two_element_repr_prime_ideal(p, hnf, field, f):
"""
return alpha such that (p, alpha) is same ideal to
the ideal represented by hnf (matrix) w.r.t. the integer ring of field.
we assume the ideal is a prime ideal whose norm is p^f, where p is prime.
"""
# assume that first basis of the integral basis equals 1
int_basis = field.integer_ring()
# step 1
n = field.degree
Z = rational.theIntegerRing
beta = (p * matrix.unitMatrix(n, Z).subMatrix(
range(1, n + 1), range(2, n + 1))).copy()
beta.extendColumn(hnf)
m = beta.column
# step 2
R = 0
P_norm = p ** f
while True:
R += 1
lmd = [R] * m
while True:
# step 3
alpha = lmd[0] * beta[1]
for j in range(1, m):
alpha += lmd[j] * beta[j + 1]
alpha = _vector_to_algnumber(int_basis * alpha, field)
n = alpha.norm() / P_norm
r = divmod(int(n), p)[1]
if r:
return alpha
# step 4
for j in range(m - 1, 0, -1):
if lmd[j] != -R:
break
lmd[j] -= 1
for i in range(j + 1, m):
lmd[i] = R
# step 5
for j in range(m):
if lmd[j] != 0:
break
else:
break
def _splitting_squarefree(p, H, base_multiply, A, gamma_mul, alpha):
"""
split squarefree part
"""
F_p = finitefield.FinitePrimeField(p)
if H == None:
n = base_multiply.row
f = n
H_new = matrix.zeroMatrix(n, 1, F_p)
else:
n = H.row
f = n - H.column
H_new = H.copy()
# step 12
one_gamma = vector.Vector([F_p.one] + [F_p.zero] *
(f - 1)) # w.r.t. gamma_1
minpoly_matrix_alpha = matrix.FieldMatrix(f, 2, [one_gamma, alpha])
ker = minpoly_matrix_alpha.kernel()
alpha_pow = alpha
while ker == None:
alpha_pow = _vect_mul_by_base_mul(alpha_pow, alpha, gamma_mul, f)
minpoly_matrix_alpha.extendColumn(alpha_pow)
ker = minpoly_matrix_alpha.kernel()
minpoly_alpha = uniutil.FinitePrimeFieldPolynomial(enumerate(ker[1].compo),
F_p)
# step 13
m_list = minpoly_alpha.factor()
# step 14
L = []
for (m_s, i) in m_list:
M_s = H_new.copy()
beta_s_gamma = _substitution_by_base_mul(m_s, alpha, gamma_mul,
one_gamma, f)
# beta_s_gamma (w.r.t. gamma) -> beta_s (w.r.t. omega)
beta_s = A * beta_s_gamma
omega_beta_s = _matrix_mul_by_base_mul(matrix.unitMatrix(n, F_p),
beta_s.toMatrix(True),
base_multiply)
M_s.extendColumn(omega_beta_s)
L.append(M_s.image())
return L
def testEasyCase(self):
# Q_rt_minus_1 [1, 0, 1]
field = algfield.NumberField(self.Q_rt_minus_1)
# p=2
Q_rt_minus_1_2_index = [(2, 1)] # ramify
Q_rt_minus_1_2_prime = 2 * module.Ideal(
matrix.unitMatrix(field.degree), field, field.integer_ring())
result_e_f, result_mul = self.result_transform(prime_decomp(2, self.Q_rt_minus_1))
self.assertEqual(result_e_f, Q_rt_minus_1_2_index)
self.assertEqual(result_mul, Q_rt_minus_1_2_prime)
# p=3
Q_rt_minus_1_3_index = [(1, 2)] # inert
Q_rt_minus_1_3_prime = 3 * module.Ideal(
matrix.unitMatrix(field.degree), field, field.integer_ring())
result_e_f, result_mul = self.result_transform(prime_decomp(3, self.Q_rt_minus_1))
self.assertEqual(result_e_f, Q_rt_minus_1_3_index)
self.assertEqual(result_mul, Q_rt_minus_1_3_prime)
# p=5
Q_rt_minus_1_5_index = [(1, 1), (1, 1)] # split
Q_rt_minus_1_5_prime = 5 * module.Ideal(
matrix.unitMatrix(field.degree), field, field.integer_ring())
result_e_f, result_mul = self.result_transform(prime_decomp(5, self.Q_rt_minus_1))
self.assertEqual(result_e_f, Q_rt_minus_1_5_index)
self.assertEqual(result_mul, Q_rt_minus_1_5_prime)
# Q_cb_2 [-2, 0, 0, 1]
field = algfield.NumberField(self.Q_cb_2)
# p=3
Q_cb_2_3_index = [(3, 1)] # ramify
Q_cb_2_3_prime = 3 * module.Ideal(
matrix.unitMatrix(field.degree), field, field.integer_ring())
result_e_f, result_mul = self.result_transform(prime_decomp(3, self.Q_cb_2))
self.assertEqual(result_e_f, Q_cb_2_3_index)
self.assertEqual(result_mul, Q_cb_2_3_prime)
# p=5
Q_cb_2_5_index = [(1, 1), (1, 2)] # split & inert
Q_cb_2_5_prime = 5 * module.Ideal(
matrix.unitMatrix(field.degree), field, field.integer_ring())
result_e_f, result_mul = self.result_transform(prime_decomp(5, self.Q_cb_2))
self.assertEqual(result_e_f, Q_cb_2_5_index)
self.assertEqual(result_mul, Q_cb_2_5_prime)
# p=31
Q_cb_2_31_index = [(1, 1), (1, 1), (1, 1)] # split
Q_cb_2_31_prime = 31 * module.Ideal(
matrix.unitMatrix(field.degree), field, field.integer_ring())
result_e_f, result_mul = self.result_transform(prime_decomp(31, self.Q_cb_2))
self.assertEqual(result_e_f, Q_cb_2_31_index)
self.assertEqual(result_mul, Q_cb_2_31_prime)
def __init__(self, pair_mat_repr, number_field, base=None, ishnf=False):
"""
pair_mat_repr: generators represented with respect to base_module
1: [[deg(number_field))-integral tuple/vector], denominator]
2: [integral matrix whose the number of row is deg(number_field),
denominator]
3: rational matrix whose the number of row is deg(number_field)
number_field: an instance of NumberField
base: a base module represented with respect to Z[theta]
1: [deg(number_field)-rational tuple/vector], which represents basis
2: deg(number_field)-square non-singular rational matrix
ishnf: if hnf_repr is already HNF-form, set True
"""
self.number_field = number_field
size = self.number_field.degree
if isinstance(base, bool):
ishnf = base
base = None
if base == None: # power basis
self.base = matrix.unitMatrix(size, rational.theIntegerRing)
self.base.toFieldMatrix()
elif isinstance(base, list): # list repr
self.base = matrix.FieldSquareMatrix(size, base)
else: # matrix repr
if base.row != size:
raise TypeError("The number of row of base is incorrect.")
self.base = base.copy()
self.base.toFieldMatrix()
if not isinstance(pair_mat_repr, list): #rational matrix repr
self.mat_repr, self.denominator = _toIntegerMatrix(
pair_mat_repr)
else:
self.denominator = pair_mat_repr[1]
if isinstance(pair_mat_repr[0], list): # list repr
column = len(pair_mat_repr[0])
self.mat_repr = Submodule(
size, column, pair_mat_repr[0], ishnf)
else: # integral matrix repr
self.mat_repr = Submodule.fromMatrix(
pair_mat_repr[0].copy(), ishnf)
def __init__(self, pair_mat_repr, number_field, base=None, ishnf=False):
"""
pair_mat_repr: generators represented with respect to base_module
1: [[deg(number_field))-integral tuple/vector], denominator]
2: [integral matrix whose the number of row is deg(number_field),
denominator]
3: rational matrix whose the number of row is deg(number_field)
number_field: an instance of NumberField
base: a base module represented with respect to Z[theta]
1: [deg(number_field)-rational tuple/vector], which represents basis
2: deg(number_field)-square non-singular rational matrix
ishnf: if hnf_repr is already HNF-form, set True
"""
self.number_field = number_field
size = self.number_field.degree
if isinstance(base, bool):
ishnf = base
base = None
if base == None: # power basis
self.base = matrix.unitMatrix(size, rational.theIntegerRing)
self.base.toFieldMatrix()
elif isinstance(base, list): # list repr
self.base = matrix.FieldSquareMatrix(size, base)
else: # matrix repr
if base.row != size:
raise TypeError("The number of row of base is incorrect.")
self.base = base.copy()
self.base.toFieldMatrix()
if not isinstance(pair_mat_repr, list): #rational matrix repr
self.mat_repr, self.denominator = _toIntegerMatrix(
pair_mat_repr)
else:
self.denominator = pair_mat_repr[1]
if isinstance(pair_mat_repr[0], list): # list repr
column = len(pair_mat_repr[0])
self.mat_repr = Submodule(
size, column, pair_mat_repr[0], ishnf)
else: # integral matrix repr
self.mat_repr = Submodule.fromMatrix(
pair_mat_repr[0].copy(), ishnf)
def _splitting_squarefree(p, H, base_multiply, A, gamma_mul, alpha):
"""
split squarefree part
"""
F_p = finitefield.FinitePrimeField(p)
if H == None:
n = base_multiply.row
f = n
H_new = matrix.zeroMatrix(n, 1, F_p)
else:
n = H.row
f = n - H.column
H_new = H.copy()
# step 12
one_gamma = vector.Vector([F_p.one] + [F_p.zero] * (f - 1)) # w.r.t. gamma_1
minpoly_matrix_alpha = matrix.FieldMatrix(f, 2, [one_gamma, alpha])
ker = minpoly_matrix_alpha.kernel()
alpha_pow = alpha
while ker == None:
alpha_pow = _vect_mul_by_base_mul(alpha_pow, alpha, gamma_mul, f)
minpoly_matrix_alpha.extendColumn(alpha_pow)
ker = minpoly_matrix_alpha.kernel()
minpoly_alpha = uniutil.FinitePrimeFieldPolynomial(
enumerate(ker[1].compo), F_p)
# step 13
m_list = minpoly_alpha.factor()
# step 14
L = []
for (m_s, i) in m_list:
M_s = H_new.copy()
beta_s_gamma = _substitution_by_base_mul(
m_s, alpha, gamma_mul, one_gamma, f)
# beta_s_gamma (w.r.t. gamma) -> beta_s (w.r.t. omega)
beta_s = A * beta_s_gamma
omega_beta_s = _matrix_mul_by_base_mul(
matrix.unitMatrix(n, F_p), beta_s.toMatrix(True), base_multiply)
M_s.extendColumn(omega_beta_s)
L.append(M_s.image())
return L
def radical(self):
p = self.field.getCharacteristic()
l = int(math.log(self.dim, p))
# B = A union {1_n}
# If we assume that the algebra is unital we don't need this
B = self.stconsts + [self.unit]
# I_-1 = A, represented by a matrix of basis vectors over A
I = matrix.unitMatrix(len(B), self.field)
I.deleteColumn(len(B))
with pool.InPool() as Pool:
for i in range1(0, l):
print >>sys.stderr, "Radical computation #%d" % i
M = matrix.Matrix(len(B), I.column, self.field)
for k in range1(I.column):
X = self.toMatrix(I[k], B)
Xp = pack_matrix(X)
work = [(p, i, Xp, pack_matrix(B[j-1])) for j in range1(len(B))]
vec = map(self.field.createElement,
Pool.map(compute_g_worker, work))
M.setColumn(k, vector.Vector(vec))
#for j in range1(len(B)):
# M[j,k] = compute_g(p, i, X*B[j-1])
basis = M.kernel()
if basis is None:
self.semisimple = True
return None
# The solution is a set of column vectors sum_j(x_j * i_(i-1),j)
# where i_k,1, ..., i_k,n = basis(I_k). Here we transform these
# back to vectors in the basis of B by multiplying by basis(I_k).
zero = vector.Vector([ self.field.zero for i in range(I.row) ])
I = matrix.Matrix(len(B), basis.column,
[ sum((basis[i,j] * I[i] for i in range1(I.column)), zero)
for j in range1(basis.column)])
assert(all(x == self.field.zero for x in I.getRow(I.row)))
self.Rad = I.getBlock(1, 1, self.dim, I.column)
self.save()
return self.Rad
def T(A, j):
p = A.coeff_ring.getCharacteristic()
phi = lambda A: (x * A + matrix.unitMatrix(A.row, A.coeff_ring)).determinant()
return phi(A)[p**j]
def POLRED(self):
"""
Given a polynomial f i.e. a field self, output some polynomials
defining subfield of self, where self is a field defined by f.
Algorithm 4.4.11 in Cohen's book.
"""
n = self.degree
appr = self.getConj()
#Step 1.
Basis = self.integer_ring()
BaseList = []
for i in range(n):
AlgInt = MatAlgNumber(Basis[i].compo, self.polynomial)
BaseList.append(AlgInt)
#Step 2.
traces = []
if self.signature()[1] == 0:
for i in range(n):
for j in range(n):
s = BaseList[i] * BaseList[j]
traces.append(s.trace())
else:
sigma = self.getConj()
f = []
for i in range(n):
f.append(zpoly(Basis[i]))
for i in range(n):
for j in range(n):
m = 0 + 0j
for k in range(n):
m += f[i](sigma[k]) * f[j](sigma[k].conjugate())
traces.append(m.real)
#Step 3.
M = matrix.createMatrix(n, n, traces)
S = matrix.unitMatrix(n)
L = lattice.LLL(S, M)[0]
#Step 4.
Ch_Basis = []
for i in range(n):
base_cor = changetype(0, self.polynomial).ch_matrix()
for v in range(n):
base_cor += BaseList[v] * L.compo[v][i]
Ch_Basis.append(base_cor)
C = []
a = Ch_Basis[0]
for i in range(n):
coeff = Ch_Basis[i].ch_basic().getCharPoly()
C.append(zpoly(coeff))
#Step 5.
P = []
for i in range(n):
diff_C = C[i].differentiate()
gcd_C = C[i].subresultant_gcd(diff_C)
P.append(C[i].exact_division(gcd_C))
return P
def POLRED(self):
"""
Given a polynomial f i.e. a field self, output some polynomials
defining subfield of self, where self is a field defined by f.
Algorithm 4.4.11 in Cohen's book.
"""
n = self.degree
appr = self.getConj()
#Step 1.
Basis = self.integer_ring()
BaseList = []
for i in range(n):
AlgInt = MatAlgNumber(Basis[i].compo, self.polynomial)
BaseList.append(AlgInt)
#Step 2.
traces = []
if self.signature()[1] == 0:
for i in range(n):
for j in range(n):
s = BaseList[i]*BaseList[j]
traces.append(s.trace())
else:
sigma = self.getConj()
f = []
for i in range(n):
f.append(zpoly(Basis[i]))
for i in range(n):
for j in range(n):
m = 0 + 0j
for k in range(n):
m += f[i](sigma[k])*f[j](sigma[k].conjugate())
traces.append(m.real)
#Step 3.
M = matrix.createMatrix(n, n, traces)
S = matrix.unitMatrix(n)
L = lattice.LLL(S, M)[0]
#Step 4.
Ch_Basis = []
for i in range(n):
base_cor = changetype(0, self.polynomial).ch_matrix()
for v in range(n):
base_cor += BaseList[v]*L.compo[v][i]
Ch_Basis.append(base_cor)
C = []
a = Ch_Basis[0]
for i in range(n):
coeff = Ch_Basis[i].ch_basic().getCharPoly()
C.append(zpoly(coeff))
#Step 5.
P = []
for i in range(n):
diff_C = C[i].differentiate()
gcd_C = C[i].subresultant_gcd(diff_C)
P.append(C[i].exact_division(gcd_C))
return P