53 KiB
53 KiB
def basis_holomorphic_differentials_degree(f, m, p):
r = f.degree()
delta = GCD(r, m)
Rx.<x> = PolynomialRing(GF(p))
Rxy.<x, y> = PolynomialRing(GF(p), 2)
Fxy = FractionField(Rxy)
#########basis of holomorphic differentials and de Rham
basis_holo = []
degrees0 = {}
k = 0
for j in range(1, m):
for i in range(1, r):
if (r*j - m*i >= delta):
basis_holo += [Fxy(x^(i-1)/y^j)]
degrees0[k] = (i-1, j)
k = k+1
return(basis_holo, degrees0)
def holomorphic_differentials_basis(f, m, p):
basis_holo, degrees0 = basis_holomorphic_differentials_degree(f, m, p)
return basis_holo
def degrees_holomorphic_differentials(f, m, p):
basis_holo, degrees0 = basis_holomorphic_differentials_degree(f, m, p)
return degrees0
def basis_de_rham_degrees(f, m, p):
r = f.degree()
delta = GCD(r, m)
Rx.<x> = PolynomialRing(GF(p))
Rxy.<x, y> = PolynomialRing(GF(p), 2)
Fxy = FractionField(Rxy)
basis_holo = holomorphic_differentials_basis(f, m, p)
basis = []
for k in range(0, len(basis_holo)):
basis += [(basis_holo[k], Rx(0))]
## non-holomorphic elts of H^1_dR
t = len(basis)
degrees0 = {}
degrees1 = {}
for j in range(1, m):
for i in range(1, r):
if (r*(m-j) - m*i >= delta):
s = Rx(m-j)*Rx(x)*Rx(f.derivative()) - Rx(m)*Rx(i)*f
psi = Rx(cut(s, i))
basis += [(Fxy(psi/y^j), Fxy(m*y^(m-j)/x^i))]
degrees0[t] = (psi.degree(), j)
degrees1[t] = (-i, m-j)
t += 1
return basis, degrees0, degrees1
def de_rham_basis(f, m, p):
basis, degrees0, degrees1 = basis_de_rham_degrees(f, m, p)
return basis
def degrees_de_rham0(f, m, p):
basis, degrees0, degrees1 = basis_de_rham_degrees(f, m, p)
return degrees0
def degrees_de_rham1(f, m, p):
basis, degrees0, degrees1 = basis_de_rham_degrees(f, m, p)
return degrees1
class superelliptic:
def __init__(self, f, m, p):
Rx.<x> = PolynomialRing(GF(p))
Rxy.<x, y> = PolynomialRing(GF(p), 2)
Fxy = FractionField(Rxy)
self.polynomial = Rx(f)
self.exponent = m
self.characteristic = p
r = Rx(f).degree()
delta = GCD(r, m)
self.degree_holo = degrees_holomorphic_differentials(f, m, p)
self.degree_de_rham0 = degrees_de_rham0(f, m, p)
self.degree_de_rham1 = degrees_de_rham1(f, m, p)
holo_basis = holomorphic_differentials_basis(f, m, p)
holo_basis_converted = []
for a in holo_basis:
holo_basis_converted += [superelliptic_form(self, a)]
self.basis_holomorphic_differentials = holo_basis_converted
dr_basis = de_rham_basis(f, m, p)
dr_basis_converted = []
for (a, b) in dr_basis:
dr_basis_converted += [superelliptic_cech(self, superelliptic_form(self, a), superelliptic_function(self, b))]
self.basis_de_rham = dr_basis_converted
def __repr__(self):
f = self.polynomial
m = self.exponent
p = self.characteristic
return 'Superelliptic curve with the equation y^' + str(m) + ' = ' + str(f)+' over finite field with ' + str(p) + ' elements.'
def is_smooth(self):
f = self.polynomial
if f.discriminant() == 0:
return 0
return 1
def genus(self):
r = self.polynomial.degree()
m = self.exponent
delta = GCD(r, m)
return 1/2*((r-1)*(m-1) - delta + 1)
def verschiebung_matrix(self):
basis = self.basis_de_rham
g = self.genus()
p = self.characteristic
M = matrix(GF(p), 2*g, 2*g)
for i in range(0, len(basis)):
w = basis[i]
v = w.verschiebung().coordinates()
M[i, :] = v
return M
def frobenius_matrix(self):
basis = self.basis_de_rham
g = self.genus()
p = self.characteristic
M = matrix(GF(p), 2*g, 2*g)
for i in range(0, len(basis)):
w = basis[i]
v = w.frobenius().coordinates()
M[i, :] = v
return M
def cartier_matrix(self):
basis = self.basis_holomorphic_differentials
g = self.genus()
p = self.characteristic
M = matrix(GF(p), g, g)
for i in range(0, len(basis)):
w = basis[i]
v = w.cartier().coordinates()
M[i, :] = v
return M
def p_rank(self):
return self.cartier_matrix().rank()
def final_type(self, test = 0):
F = self.frobenius_matrix()
V = self.verschiebung_matrix()
p = self.characteristic
return flag(F, V, p, test)
def reduction(C, g):
p = C.characteristic
Rxy.<x, y> = PolynomialRing(GF(p), 2)
Fxy = FractionField(Rxy)
f = C.polynomial
r = f.degree()
m = C.exponent
g = Fxy(g)
g1 = g.numerator()
g2 = g.denominator()
Rx.<x> = PolynomialRing(GF(p))
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
(A, B, C) = xgcd(FxRy(g2), FxRy(y^m - f))
g = FxRy(g1*B/A)
while(g.degree(Rxy(y)) >= m):
d = g.degree(Rxy(y))
G = coff(g, d)
i = floor(d/m)
g = g - G*y^d + f^i * y^(d%m) *G
return(FxRy(g))
def reduction_form(C, g):
p = C.characteristic
Rxy.<x, y> = PolynomialRing(GF(p), 2)
Fxy = FractionField(Rxy)
f = C.polynomial
r = f.degree()
m = C.exponent
g = reduction(C, g)
g1 = Rxy(0)
Rx.<x> = PolynomialRing(GF(p))
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
g = FxRy(g)
for j in range(0, m):
if j==0:
G = coff(g, 0)
g1 += FxRy(G)
else:
G = coff(g, j)
g1 += Fxy(y^(j-m)*f*G)
return(g1)
class superelliptic_function:
def __init__(self, C, g):
p = C.characteristic
Rxy.<x, y> = PolynomialRing(GF(p), 2)
Fxy = FractionField(Rxy)
f = C.polynomial
r = f.degree()
m = C.exponent
self.curve = C
g = reduction(C, g)
self.function = g
def __repr__(self):
return str(self.function)
def jth_component(self, j):
g = self.function
C = self.curve
p = C.characteristic
Rx.<x> = PolynomialRing(GF(p))
Fx.<x> = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
g = FxRy(g)
return coff(g, j)
def __add__(self, other):
C = self.curve
g1 = self.function
g2 = other.function
g = reduction(C, g1 + g2)
return superelliptic_function(C, g)
def __sub__(self, other):
C = self.curve
g1 = self.function
g2 = other.function
g = reduction(C, g1 - g2)
return superelliptic_function(C, g)
def __mul__(self, other):
C = self.curve
g1 = self.function
g2 = other.function
g = reduction(C, g1 * g2)
return superelliptic_function(C, g)
def __truediv__(self, other):
C = self.curve
g1 = self.function
g2 = other.function
g = reduction(C, g1 / g2)
return superelliptic_function(C, g)
def diffn(self):
C = self.curve
f = C.polynomial
m = C.exponent
p = C.characteristic
g = self.function
Rxy.<x, y> = PolynomialRing(GF(p), 2)
Fxy = FractionField(Rxy)
g = Fxy(g)
A = g.derivative(x)
B = g.derivative(y)*f.derivative(x)/(m*y^(m-1))
return superelliptic_form(C, A+B)
class superelliptic_form:
def __init__(self, C, g):
p = C.characteristic
Rxy.<x, y> = PolynomialRing(GF(p), 2)
Fxy = FractionField(Rxy)
g = Fxy(reduction_form(C, g))
self.form = g
self.curve = C
def __add__(self, other):
C = self.curve
g1 = self.form
g2 = other.form
g = reduction(C, g1 + g2)
return superelliptic_form(C, g)
def __sub__(self, other):
C = self.curve
g1 = self.form
g2 = other.form
g = reduction(C, g1 - g2)
return superelliptic_form(C, g)
def __repr__(self):
g = self.form
if len(str(g)) == 1:
return str(g) + ' dx'
return '('+str(g) + ') dx'
def __rmul__(self, constant):
C = self.curve
omega = self.form
return superelliptic_form(C, constant*omega)
def cartier(self):
C = self.curve
m = C.exponent
p = C.characteristic
f = C.polynomial
Rx.<x> = PolynomialRing(GF(p))
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
Fxy = FractionField(FxRy)
result = superelliptic_form(C, FxRy(0))
mult_order = Integers(m)(p).multiplicative_order()
M = Integer((p^(mult_order)-1)/m)
for j in range(1, m):
fct_j = self.jth_component(j)
h = Rx(fct_j*f^(M*j))
j1 = (p^(mult_order-1)*j)%m
B = floor(p^(mult_order-1)*j/m)
result += superelliptic_form(C, polynomial_part(p, h)/(f^B*y^(j1)))
return result
def coordinates(self):
C = self.curve
p = C.characteristic
m = C.exponent
Rx.<x> = PolynomialRing(GF(p))
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
g = C.genus()
degrees_holo = C.degree_holo
degrees_holo_inv = {b:a for a, b in degrees_holo.items()}
basis = C.basis_holomorphic_differentials
for j in range(1, m):
omega_j = Fx(self.jth_component(j))
if omega_j != Fx(0):
d = degree_of_rational_fctn(omega_j, p)
index = degrees_holo_inv[(d, j)]
a = coeff_of_rational_fctn(omega_j, p)
a1 = coeff_of_rational_fctn(basis[index].jth_component(j), p)
elt = self - (a/a1)*basis[index]
return elt.coordinates() + a/a1*vector([GF(p)(i == index) for i in range(0, g)])
return vector(g*[0])
def jth_component(self, j):
g = self.form
C = self.curve
p = C.characteristic
Rx.<x> = PolynomialRing(GF(p))
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
Fxy = FractionField(FxRy)
Ryinv.<y_inv> = PolynomialRing(Fx)
g = Fxy(g)
g = g(y = 1/y_inv)
g = Ryinv(g)
return coff(g, j)
def is_regular_on_U0(self):
C = self.curve
p = C.characteristic
m = C.exponent
Rx.<x> = PolynomialRing(GF(p))
for j in range(1, m):
if self.jth_component(j) not in Rx:
return 0
return 1
def is_regular_on_Uinfty(self):
C = self.curve
p = C.characteristic
m = C.exponent
f = C.polynomial
r = f.degree()
delta = GCD(m, r)
M = m/delta
R = r/delta
for j in range(1, m):
A = self.jth_component(j)
d = degree_of_rational_fctn(A, p)
if(-d*M + j*R -(M+1)<0):
return 0
return 1
class superelliptic_cech:
def __init__(self, C, omega, fct):
self.omega0 = omega
self.omega8 = omega - diffn(fct)
self.f = fct
self.curve = C
def __add__(self, other):
C = self.curve
return superelliptic_cech(C, self.omega0 + other.omega0, self.f + other.f)
def __sub__(self, other):
C = self.curve
return superelliptic_cech(C, self.omega0 - other.omega0, self.f - other.f)
def __rmul__(self, constant):
C = self.curve
w1 = self.omega0.form
f1 = self.f.function
w2 = superelliptic_form(C, constant*w1)
f2 = superelliptic_function(C, constant*f1)
return superelliptic_cech(C, w2, f2)
def __repr__(self):
return "(" + str(self.omega0) + ", " + str(self.f) + ", " + str(self.omega8) + ")"
def verschiebung(self):
C = self.curve
omega = self.omega0
p = C.characteristic
Rx.<x> = PolynomialRing(GF(p))
return superelliptic_cech(C, omega.cartier(), superelliptic_function(C, Rx(0)))
def frobenius(self):
C = self.curve
fct = self.f.function
p = C.characteristic
Rx.<x> = PolynomialRing(GF(p))
return superelliptic_cech(C, superelliptic_form(C, Rx(0)), superelliptic_function(C, fct^p))
def coordinates(self):
C = self.curve
p = C.characteristic
m = C.exponent
Rx.<x> = PolynomialRing(GF(p))
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
g = C.genus()
degrees_holo = C.degree_holo
degrees_holo_inv = {b:a for a, b in degrees_holo.items()}
degrees0 = C.degree_de_rham0
degrees0_inv = {b:a for a, b in degrees0.items()}
degrees1 = C.degree_de_rham1
degrees1_inv = {b:a for a, b in degrees1.items()}
basis = C.basis_de_rham
omega = self.omega0
fct = self.f
if fct.function == Rx(0) and omega.form != Rx(0):
for j in range(1, m):
omega_j = Fx(omega.jth_component(j))
if omega_j != Fx(0):
d = degree_of_rational_fctn(omega_j, p)
index = degrees_holo_inv[(d, j)]
a = coeff_of_rational_fctn(omega_j, p)
a1 = coeff_of_rational_fctn(basis[index].omega0.jth_component(j), p)
elt = self - (a/a1)*basis[index]
return elt.coordinates() + a/a1*vector([GF(p)(i == index) for i in range(0, 2*g)])
for j in range(1, m):
fct_j = Fx(fct.jth_component(j))
if (fct_j != Rx(0)):
d = degree_of_rational_fctn(fct_j, p)
if (d, j) in degrees1.values():
index = degrees1_inv[(d, j)]
a = coeff_of_rational_fctn(fct_j, p)
elt = self - (a/m)*basis[index]
return elt.coordinates() + a/m*vector([GF(p)(i == index) for i in range(0, 2*g)])
if d<0:
a = coeff_of_rational_fctn(fct_j, p)
h = superelliptic_function(C, FxRy(a*y^j*x^d))
elt = superelliptic_cech(C, self.omega0, self.f - h)
return elt.coordinates()
if (fct_j != Rx(0)):
G = superelliptic_function(C, y^j*x^d)
a = coeff_of_rational_fctn(fct_j, p)
elt =self - a*superelliptic_cech(C, diffn(G), G)
return elt.coordinates()
return vector(2*g*[0])
def is_cocycle(self):
w0 = self.omega0
w8 = self.omega8
fct = self.f
if not w0.is_regular_on_U0() and not w8.is_regular_on_Uinfty():
return('w0 & w8')
if not w0.is_regular_on_U0():
return('w0')
if not w8.is_regular_on_Uinfty():
return('w8')
if w0.is_regular_on_U0() and w8.is_regular_on_Uinfty():
return 1
return 0
def degree_of_rational_fctn(f, p):
Rx.<x> = PolynomialRing(GF(p))
Fx = FractionField(Rx)
f = Fx(f)
f1 = f.numerator()
f2 = f.denominator()
d1 = f1.degree()
d2 = f2.degree()
return(d1 - d2)
def coeff_of_rational_fctn(f, p):
Rx.<x> = PolynomialRing(GF(p))
Fx = FractionField(Rx)
f = Fx(f)
if f == Rx(0):
return 0
f1 = f.numerator()
f2 = f.denominator()
d1 = f1.degree()
d2 = f2.degree()
a1 = f1.coefficients(sparse = false)[d1]
a2 = f2.coefficients(sparse = false)[d2]
return(a1/a2)
def coff(f, d):
lista = f.coefficients(sparse = false)
if len(lista) <= d:
return 0
return lista[d]
def cut(f, i):
R = f.parent()
coeff = f.coefficients(sparse = false)
return sum(R(x^(j-i-1)) * coeff[j] for j in range(i+1, f.degree() + 1))
def polynomial_part(p, h):
Rx.<x> = PolynomialRing(GF(p))
h = Rx(h)
result = Rx(0)
for i in range(0, h.degree()+1):
if (i%p) == p-1:
power = Integer((i-(p-1))/p)
result += Integer(h[i]) * x^(power)
return resultdef preimage(U, V, M): #preimage of subspace U under M
basis_preimage = M.right_kernel().basis()
imageU = U.intersection(M.transpose().image())
basis = imageU.basis()
for v in basis:
w = M.solve_right(v)
basis_preimage = basis_preimage + [w]
return V.subspace(basis_preimage)
def image(U, V, M):
basis = U.basis()
basis_image = []
for v in basis:
basis_image += [M*v]
return V.subspace(basis_image)
def flag(F, V, p, test = 0):
dim = F.dimensions()[0]
space = VectorSpace(GF(p), dim)
flag_subspaces = (dim+1)*[0]
flag_used = (dim+1)*[0]
final_type = (dim+1)*['?']
flag_subspaces[dim] = space
flag_used[dim] = 1
while 1 in flag_used:
index = flag_used.index(1)
flag_used[index] = 0
U = flag_subspaces[index]
U_im = image(U, space, V)
d_im = U_im.dimension()
final_type[index] = d_im
U_pre = preimage(U, space, F)
d_pre = U_pre.dimension()
if flag_subspaces[d_im] == 0:
flag_subspaces[d_im] = U_im
flag_used[d_im] = 1
if flag_subspaces[d_pre] == 0:
flag_subspaces[d_pre] = U_pre
flag_used[d_pre] = 1
if test == 1:
print('(', final_type, ')')
for i in range(0, dim+1):
if final_type[i] == '?' and final_type[dim - i] != '?':
i1 = dim - i
final_type[i] = final_type[i1] - i1 + dim/2
final_type[0] = 0
for i in range(1, dim+1):
if final_type[i] == '?':
prev = final_type[i-1]
if prev != '?' and prev in final_type[i+1:]:
final_type[i] = prev
for i in range(1, dim+1):
if final_type[i] == '?':
final_type[i] = min(final_type[i-1] + 1, dim/2)
if is_final(final_type, dim/2):
return final_type[1:dim/2 + 1]
print('error:', final_type[1:dim/2 + 1])
def is_final(final_type, dim):
n = len(final_type)
if final_type[0] != 0:
return 0
if final_type[n-1] != dim:
return 0
for i in range(1, n):
if final_type[i] != final_type[i - 1] and final_type[i] != final_type[i - 1] + 1:
return 0
return 1def dzialanie(f, m, p):
Rx.<x> = PolynomialRing(GF(p))
fp = Rx(f(x^p))
C = superelliptic(fp, m, p)
holo = C.basis_holomorphic_differentials
Rxy.<x, y> = PolynomialRing(GF(p), 2)
kxi1.<xi1> = PolynomialRing(FractionField(Rxy))
kxi = kxi1.quotient(xi1^p)
xi = kxi(xi1)
holo_forms = [a.form for a in holo]
holo_xi = [kxi(a(x = x+xi, y = y)) for a in holo_forms]
N = matrix(kxi1, C.genus(), C.genus())
for i in range(0, p):
M = matrix(GF(p), C.genus(), C.genus())
for j in range(0, len(holo_xi)):
a = holo_xi[j]
omega = superelliptic_form(C, a[i])
v = omega.coordinates()
M[j, :] = v
N += xi1^i*M
return N
def bloki(A):
B = A.jordan_form(subdivide=True)
lista = []
d = 0
i = 0
while d < B.dimensions()[1]:
lista.append(B.subdivision(i, i).dimensions()[1])
d = d + B.subdivision(i, i).dimensions()[1]
i = i+1
return lista
def p_cov(C):
m = C.exponent
p = C.characteristic
f = C.polynomial
return superelliptic(f(x^p), m, p)Testy
p = 5
Rx.<x> = PolynomialRing(GF(p))
r = 3
f = x^(r) + x + 1
m = 12
A = dzialanie(f, m, p)
print(bloki(A))
C = superelliptic(f, m, p)
print(C.genus(), p_cov(C).genus())[5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4, 3, 3, 3, 2, 2, 2, 1, 1, 1] 10 76
omega = C.basis_holomorphic_differentials[2]lista = [a.form for a in C.basis_holomorphic_differentials]kxi1.<xi1> = PolynomialRing(FractionField(Rxy))
kxi = kxi1.quotient(xi1^p)
xi = kxi(xi1)lista2 = [kxi(a(x = x+xi, y = y)) for a in lista]kxi(x)x
a = lista2[0]omega.form(x = x+xi, y = y)1/y*xi1bar^2 + 2*x/y*xi1bar + x^2/y
omega1 = superelliptic_form(C, omega.form(x = x+xi, y = y))[0;31m---------------------------------------------------------------------------[0m [0;31mAttributeError[0m Traceback (most recent call last) [0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/rings/fraction_field.py[0m in [0;36m_element_constructor_[0;34m(self, x, y, coerce)[0m [1;32m 695[0m [0;32mtry[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [0;32m--> 696[0;31m [0mx[0m[0;34m,[0m [0my[0m [0;34m=[0m [0mresolve_fractions[0m[0;34m([0m[0mx0[0m[0;34m,[0m [0my0[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 697[0m [0;32mexcept[0m [0;34m([0m[0mAttributeError[0m[0;34m,[0m [0mTypeError[0m[0;34m)[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/rings/fraction_field.py[0m in [0;36mresolve_fractions[0;34m(x, y)[0m [1;32m 672[0m [0;32mdef[0m [0mresolve_fractions[0m[0;34m([0m[0mx[0m[0;34m,[0m [0my[0m[0;34m)[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [0;32m--> 673[0;31m [0mxn[0m [0;34m=[0m [0mx[0m[0;34m.[0m[0mnumerator[0m[0;34m([0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 674[0m [0mxd[0m [0;34m=[0m [0mx[0m[0;34m.[0m[0mdenominator[0m[0;34m([0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/structure/element.pyx[0m in [0;36msage.structure.element.Element.__getattr__ (build/cythonized/sage/structure/element.c:4754)[0;34m()[0m [1;32m 493[0m """ [0;32m--> 494[0;31m [0;32mreturn[0m [0mself[0m[0;34m.[0m[0mgetattr_from_category[0m[0;34m([0m[0mname[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 495[0m [0;34m[0m[0m [0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/structure/element.pyx[0m in [0;36msage.structure.element.Element.getattr_from_category (build/cythonized/sage/structure/element.c:4866)[0;34m()[0m [1;32m 506[0m [0mcls[0m [0;34m=[0m [0mP[0m[0;34m.[0m[0m_abstract_element_class[0m[0;34m[0m[0;34m[0m[0m [0;32m--> 507[0;31m [0;32mreturn[0m [0mgetattr_from_other_class[0m[0;34m([0m[0mself[0m[0;34m,[0m [0mcls[0m[0;34m,[0m [0mname[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 508[0m [0;34m[0m[0m [0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/cpython/getattr.pyx[0m in [0;36msage.cpython.getattr.getattr_from_other_class (build/cythonized/sage/cpython/getattr.c:2566)[0;34m()[0m [1;32m 355[0m [0mdummy_error_message[0m[0;34m.[0m[0mname[0m [0;34m=[0m [0mname[0m[0;34m[0m[0;34m[0m[0m [0;32m--> 356[0;31m [0;32mraise[0m [0mAttributeError[0m[0;34m([0m[0mdummy_error_message[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 357[0m [0mcdef[0m [0mPyObject[0m[0;34m*[0m [0mattr[0m [0;34m=[0m [0minstance_getattr[0m[0;34m([0m[0mcls[0m[0;34m,[0m [0mname[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0;31mAttributeError[0m: 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomialRing_libsingular' object has no attribute '__custom_name' During handling of the above exception, another exception occurred: [0;31mTypeError[0m Traceback (most recent call last) [0;32m/tmp/ipykernel_1899/916120484.py[0m in [0;36m<module>[0;34m[0m [0;32m----> 1[0;31m [0momega1[0m [0;34m=[0m [0msuperelliptic_form[0m[0;34m([0m[0mC[0m[0;34m,[0m [0momega[0m[0;34m.[0m[0mform[0m[0;34m([0m[0mx[0m [0;34m=[0m [0mx[0m[0;34m+[0m[0mxi[0m[0;34m,[0m [0my[0m [0;34m=[0m [0my[0m[0;34m)[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m [0;32m/tmp/ipykernel_1899/3188561669.py[0m in [0;36m__init__[0;34m(self, C, g)[0m [1;32m 283[0m [0mRxy[0m [0;34m=[0m [0mPolynomialRing[0m[0;34m([0m[0mGF[0m[0;34m([0m[0mp[0m[0;34m)[0m[0;34m,[0m [0mInteger[0m[0;34m([0m[0;36m2[0m[0;34m)[0m[0;34m,[0m [0mnames[0m[0;34m=[0m[0;34m([0m[0;34m'x'[0m[0;34m,[0m [0;34m'y'[0m[0;34m,[0m[0;34m)[0m[0;34m)[0m[0;34m;[0m [0;34m([0m[0mx[0m[0;34m,[0m [0my[0m[0;34m,[0m[0;34m)[0m [0;34m=[0m [0mRxy[0m[0;34m.[0m[0m_first_ngens[0m[0;34m([0m[0;36m2[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [1;32m 284[0m [0mFxy[0m [0;34m=[0m [0mFractionField[0m[0;34m([0m[0mRxy[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0;32m--> 285[0;31m [0mg[0m [0;34m=[0m [0mFxy[0m[0;34m([0m[0mreduction_form[0m[0;34m([0m[0mC[0m[0;34m,[0m [0mg[0m[0;34m)[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 286[0m [0mself[0m[0;34m.[0m[0mform[0m [0;34m=[0m [0mg[0m[0;34m[0m[0;34m[0m[0m [1;32m 287[0m [0mself[0m[0;34m.[0m[0mcurve[0m [0;34m=[0m [0mC[0m[0;34m[0m[0;34m[0m[0m [0;32m/tmp/ipykernel_1899/3188561669.py[0m in [0;36mreduction_form[0;34m(C, g)[0m [1;32m 194[0m [0mr[0m [0;34m=[0m [0mf[0m[0;34m.[0m[0mdegree[0m[0;34m([0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [1;32m 195[0m [0mm[0m [0;34m=[0m [0mC[0m[0;34m.[0m[0mexponent[0m[0;34m[0m[0;34m[0m[0m [0;32m--> 196[0;31m [0mg[0m [0;34m=[0m [0mreduction[0m[0;34m([0m[0mC[0m[0;34m,[0m [0mg[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 197[0m [0;34m[0m[0m [1;32m 198[0m [0mg1[0m [0;34m=[0m [0mRxy[0m[0;34m([0m[0mInteger[0m[0;34m([0m[0;36m0[0m[0;34m)[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0;32m/tmp/ipykernel_1899/3188561669.py[0m in [0;36mreduction[0;34m(C, g)[0m [1;32m 169[0m [0mr[0m [0;34m=[0m [0mf[0m[0;34m.[0m[0mdegree[0m[0;34m([0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [1;32m 170[0m [0mm[0m [0;34m=[0m [0mC[0m[0;34m.[0m[0mexponent[0m[0;34m[0m[0;34m[0m[0m [0;32m--> 171[0;31m [0mg[0m [0;34m=[0m [0mFxy[0m[0;34m([0m[0mg[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 172[0m [0mg1[0m [0;34m=[0m [0mg[0m[0;34m.[0m[0mnumerator[0m[0;34m([0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [1;32m 173[0m [0mg2[0m [0;34m=[0m [0mg[0m[0;34m.[0m[0mdenominator[0m[0;34m([0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/structure/parent.pyx[0m in [0;36msage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9388)[0;34m()[0m [1;32m 896[0m [0;32mif[0m [0mmor[0m [0;32mis[0m [0;32mnot[0m [0;32mNone[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [1;32m 897[0m [0;32mif[0m [0mno_extra_args[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [0;32m--> 898[0;31m [0;32mreturn[0m [0mmor[0m[0;34m.[0m[0m_call_[0m[0;34m([0m[0mx[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 899[0m [0;32melse[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [1;32m 900[0m [0;32mreturn[0m [0mmor[0m[0;34m.[0m[0m_call_with_args[0m[0;34m([0m[0mx[0m[0;34m,[0m [0margs[0m[0;34m,[0m [0mkwds[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/structure/coerce_maps.pyx[0m in [0;36msage.structure.coerce_maps.DefaultConvertMap_unique._call_ (build/cythonized/sage/structure/coerce_maps.c:4665)[0;34m()[0m [1;32m 159[0m [0mprint[0m[0;34m([0m[0mtype[0m[0;34m([0m[0mC[0m[0;34m)[0m[0;34m,[0m [0mC[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [1;32m 160[0m [0mprint[0m[0;34m([0m[0mtype[0m[0;34m([0m[0mC[0m[0;34m.[0m[0m_element_constructor[0m[0;34m)[0m[0;34m,[0m [0mC[0m[0;34m.[0m[0m_element_constructor[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0;32m--> 161[0;31m [0;32mraise[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 162[0m [0;34m[0m[0m [1;32m 163[0m [0mcpdef[0m [0mElement[0m [0m_call_with_args[0m[0;34m([0m[0mself[0m[0;34m,[0m [0mx[0m[0;34m,[0m [0margs[0m[0;34m=[0m[0;34m([0m[0;34m)[0m[0;34m,[0m [0mkwds[0m[0;34m=[0m[0;34m{[0m[0;34m}[0m[0;34m)[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/structure/coerce_maps.pyx[0m in [0;36msage.structure.coerce_maps.DefaultConvertMap_unique._call_ (build/cythonized/sage/structure/coerce_maps.c:4557)[0;34m()[0m [1;32m 154[0m [0mcdef[0m [0mParent[0m [0mC[0m [0;34m=[0m [0mself[0m[0;34m.[0m[0m_codomain[0m[0;34m[0m[0;34m[0m[0m [1;32m 155[0m [0;32mtry[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [0;32m--> 156[0;31m [0;32mreturn[0m [0mC[0m[0;34m.[0m[0m_element_constructor[0m[0;34m([0m[0mx[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 157[0m [0;32mexcept[0m [0mException[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [1;32m 158[0m [0;32mif[0m [0mprint_warnings[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/rings/fraction_field.py[0m in [0;36m_element_constructor_[0;34m(self, x, y, coerce)[0m [1;32m 696[0m [0mx[0m[0;34m,[0m [0my[0m [0;34m=[0m [0mresolve_fractions[0m[0;34m([0m[0mx0[0m[0;34m,[0m [0my0[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [1;32m 697[0m [0;32mexcept[0m [0;34m([0m[0mAttributeError[0m[0;34m,[0m [0mTypeError[0m[0;34m)[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [0;32m--> 698[0;31m raise TypeError("cannot convert {!r}/{!r} to an element of {}".format( [0m[1;32m 699[0m x0, y0, self)) [1;32m 700[0m [0;32mtry[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [0;31mTypeError[0m: cannot convert 1/y*xi1bar^2 + 2*x/y*xi1bar + x^2/y/1 to an element of Fraction Field of Multivariate Polynomial Ring in x, y over Finite Field of size 5
kxi1.<xi> = PolynomialRing(GF(p))
kxi = FractionField(kxi1)
Rxy.<x, y> = PolynomialRing(kxi, 2)
Fxy = FractionField(Rxy)
Rx.<x> = PolynomialRing(kxi)
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)C.basis_holomorphic_differentialsx/y
p = 5
Rx.<x> = PolynomialRing(GF(p))
f = Rx(x^7 + x + 1)
m = 2
Rxy.<x, y> = PolynomialRing(GF(p), 2)
print(dzialanie(f, m, p))
A = dzialanie(f, m, p)[ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [ xi1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [ xi1^2 2*xi1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [ xi1^3 -2*xi1^2 -2*xi1 1 0 0 0 0 0 0 0 0 0 0 0 0 0] [ xi1^4 -xi1^3 xi1^2 -xi1 1 0 0 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 0 0 xi1 1 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 0 0 xi1^2 2*xi1 1 0 0 0 0 0 0 0 0 0] [ 0 0 0 0 0 xi1^3 -2*xi1^2 -2*xi1 1 0 0 0 0 0 0 0 0] [ 0 0 0 0 0 xi1^4 -xi1^3 xi1^2 -xi1 1 0 0 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 xi1 1 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 xi1^2 2*xi1 1 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 xi1^3 -2*xi1^2 -2*xi1 1 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 xi1^4 -xi1^3 xi1^2 -xi1 1 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 xi1 1]
A.jordan_form()[1 1 0 0 0|0 0 0 0 0|0 0 0 0 0|0 0] [0 1 1 0 0|0 0 0 0 0|0 0 0 0 0|0 0] [0 0 1 1 0|0 0 0 0 0|0 0 0 0 0|0 0] [0 0 0 1 1|0 0 0 0 0|0 0 0 0 0|0 0] [0 0 0 0 1|0 0 0 0 0|0 0 0 0 0|0 0] [---------+---------+---------+---] [0 0 0 0 0|1 1 0 0 0|0 0 0 0 0|0 0] [0 0 0 0 0|0 1 1 0 0|0 0 0 0 0|0 0] [0 0 0 0 0|0 0 1 1 0|0 0 0 0 0|0 0] [0 0 0 0 0|0 0 0 1 1|0 0 0 0 0|0 0] [0 0 0 0 0|0 0 0 0 1|0 0 0 0 0|0 0] [---------+---------+---------+---] [0 0 0 0 0|0 0 0 0 0|1 1 0 0 0|0 0] [0 0 0 0 0|0 0 0 0 0|0 1 1 0 0|0 0] [0 0 0 0 0|0 0 0 0 0|0 0 1 1 0|0 0] [0 0 0 0 0|0 0 0 0 0|0 0 0 1 1|0 0] [0 0 0 0 0|0 0 0 0 0|0 0 0 0 1|0 0] [---------+---------+---------+---] [0 0 0 0 0|0 0 0 0 0|0 0 0 0 0|1 1] [0 0 0 0 0|0 0 0 0 0|0 0 0 0 0|0 1]
M = matrix(Rx, 3,3)M = matrix(Rx, [[x, 1, 1], [1,2,3], [x+1,2,4]])M.rank()3
kxi1.<xi1> = PolynomialRing(FractionField(Rxy))
kxi = kxi1.quotient(xi1^p)
xi = kxi(xi1)lift(xi^5)0
E = EllipticCurve(GF(3), [1,1])p = 5
R.<x> = PolynomialRing(GF(p))
f = x^3 + x + 1
m = 2
C = superelliptic(f, m, p)
C.verschiebung_matrix()[2 0] [2 0]