{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "def basis_holomorphic_differentials_degree(f, m, p):\n", " r = f.degree()\n", " delta = GCD(r, m)\n", " Rx. = PolynomialRing(GF(p))\n", " Rxy. = PolynomialRing(GF(p), 2)\n", " Fxy = FractionField(Rxy)\n", " #########basis of holomorphic differentials and de Rham\n", " \n", " basis_holo = []\n", " degrees0 = {}\n", " k = 0\n", " \n", " for j in range(1, m):\n", " for i in range(1, r):\n", " if (r*j - m*i >= delta):\n", " basis_holo += [Fxy(x^(i-1)/y^j)]\n", " degrees0[k] = (i-1, j)\n", " k = k+1\n", " \n", " return(basis_holo, degrees0)\n", "\n", "def holomorphic_differentials_basis(f, m, p):\n", " basis_holo, degrees0 = basis_holomorphic_differentials_degree(f, m, p)\n", " return basis_holo\n", " \n", "def degrees_holomorphic_differentials(f, m, p):\n", " basis_holo, degrees0 = basis_holomorphic_differentials_degree(f, m, p)\n", " return degrees0\n", " \n", "def basis_de_rham_degrees(f, m, p):\n", " r = f.degree()\n", " delta = GCD(r, m)\n", " Rx. = PolynomialRing(GF(p))\n", " Rxy. = PolynomialRing(GF(p), 2)\n", " Fxy = FractionField(Rxy)\n", " basis_holo = holomorphic_differentials_basis(f, m, p)\n", " basis = []\n", " for k in range(0, len(basis_holo)):\n", " basis += [(basis_holo[k], Rx(0))]\n", "\n", " ## non-holomorphic elts of H^1_dR\n", " t = len(basis)\n", " degrees0 = {}\n", " degrees1 = {}\n", " for j in range(1, m):\n", " for i in range(1, r):\n", " if (r*(m-j) - m*i >= delta): \n", " s = Rx(m-j)*Rx(x)*Rx(f.derivative()) - Rx(m)*Rx(i)*f\n", " psi = Rx(cut(s, i))\n", " basis += [(Fxy(psi/y^j), Fxy(m*y^(m-j)/x^i))]\n", " degrees0[t] = (psi.degree(), j)\n", " degrees1[t] = (-i, m-j)\n", " t += 1\n", " return basis, degrees0, degrees1\n", "\n", "def de_rham_basis(f, m, p):\n", " basis, degrees0, degrees1 = basis_de_rham_degrees(f, m, p)\n", " return basis\n", "\n", "def degrees_de_rham0(f, m, p):\n", " basis, degrees0, degrees1 = basis_de_rham_degrees(f, m, p)\n", " return degrees0\n", "\n", "def degrees_de_rham1(f, m, p):\n", " basis, degrees0, degrees1 = basis_de_rham_degrees(f, m, p)\n", " return degrees1 \n", "\n", "\n", "class superelliptic:\n", " \n", " def __init__(self, f, m, p):\n", " Rx. = PolynomialRing(GF(p))\n", " Rxy. = PolynomialRing(GF(p), 2)\n", " Fxy = FractionField(Rxy)\n", " self.polynomial = Rx(f)\n", " self.exponent = m\n", " self.characteristic = p\n", " \n", " r = Rx(f).degree()\n", " delta = GCD(r, m)\n", " self.degree_holo = degrees_holomorphic_differentials(f, m, p)\n", " self.degree_de_rham0 = degrees_de_rham0(f, m, p)\n", " self.degree_de_rham1 = degrees_de_rham1(f, m, p)\n", " \n", " holo_basis = holomorphic_differentials_basis(f, m, p)\n", " holo_basis_converted = []\n", " for a in holo_basis:\n", " holo_basis_converted += [superelliptic_form(self, a)]\n", " \n", " self.basis_holomorphic_differentials = holo_basis_converted\n", " \n", "\n", " dr_basis = de_rham_basis(f, m, p)\n", " dr_basis_converted = []\n", " for (a, b) in dr_basis:\n", " dr_basis_converted += [superelliptic_cech(self, superelliptic_form(self, a), superelliptic_function(self, b))]\n", " \n", " self.basis_de_rham = dr_basis_converted\n", " \n", " def __repr__(self):\n", " f = self.polynomial\n", " m = self.exponent\n", " p = self.characteristic\n", " return 'Superelliptic curve with the equation y^' + str(m) + ' = ' + str(f)+' over finite field with ' + str(p) + ' elements.'\n", " \n", " def is_smooth(self):\n", " f = self.polynomial\n", " if f.discriminant() == 0:\n", " return 0\n", " return 1\n", " \n", " def genus(self):\n", " r = self.polynomial.degree()\n", " m = self.exponent\n", " delta = GCD(r, m)\n", " return 1/2*((r-1)*(m-1) - delta + 1)\n", " \n", " def verschiebung_matrix(self):\n", " basis = self.basis_de_rham\n", " g = self.genus()\n", " p = self.characteristic\n", " M = matrix(GF(p), 2*g, 2*g)\n", " for i in range(0, len(basis)):\n", " w = basis[i]\n", " v = w.verschiebung().coordinates()\n", " M[i, :] = v\n", " return M\n", " \n", " def frobenius_matrix(self):\n", " basis = self.basis_de_rham\n", " g = self.genus()\n", " p = self.characteristic\n", " M = matrix(GF(p), 2*g, 2*g)\n", " \n", " for i in range(0, len(basis)):\n", " w = basis[i]\n", " v = w.frobenius().coordinates()\n", " M[i, :] = v\n", " return M\n", "\n", " def cartier_matrix(self):\n", " basis = self.basis_holomorphic_differentials\n", " g = self.genus()\n", " p = self.characteristic\n", " M = matrix(GF(p), g, g)\n", " for i in range(0, len(basis)):\n", " w = basis[i]\n", " v = w.cartier().coordinates()\n", " M[i, :] = v\n", " return M \n", " \n", " def p_rank(self):\n", " return self.cartier_matrix().rank()\n", " \n", " def final_type(self, test = 0):\n", " F = self.frobenius_matrix()\n", " V = self.verschiebung_matrix()\n", " p = self.characteristic\n", " return flag(F, V, p, test)\n", " \n", "def reduction(C, g):\n", " p = C.characteristic\n", " Rxy. = PolynomialRing(GF(p), 2)\n", " Fxy = FractionField(Rxy)\n", " f = C.polynomial\n", " r = f.degree()\n", " m = C.exponent\n", " g = Fxy(g)\n", " g1 = g.numerator()\n", " g2 = g.denominator()\n", " \n", " Rx. = PolynomialRing(GF(p))\n", " Fx = FractionField(Rx)\n", " FxRy. = PolynomialRing(Fx) \n", " (A, B, C) = xgcd(FxRy(g2), FxRy(y^m - f))\n", " g = FxRy(g1*B/A)\n", " \n", " while(g.degree(Rxy(y)) >= m):\n", " d = g.degree(Rxy(y))\n", " G = coff(g, d)\n", " i = floor(d/m)\n", " g = g - G*y^d + f^i * y^(d%m) *G\n", " \n", " return(FxRy(g))\n", "\n", "def reduction_form(C, g):\n", " p = C.characteristic\n", " Rxy. = PolynomialRing(GF(p), 2)\n", " Fxy = FractionField(Rxy)\n", " f = C.polynomial\n", " r = f.degree()\n", " m = C.exponent\n", " g = reduction(C, g)\n", "\n", " g1 = Rxy(0)\n", " Rx. = PolynomialRing(GF(p))\n", " Fx = FractionField(Rx)\n", " FxRy. = PolynomialRing(Fx)\n", " \n", " g = FxRy(g)\n", " for j in range(0, m):\n", " if j==0:\n", " G = coff(g, 0)\n", " g1 += FxRy(G)\n", " else:\n", " G = coff(g, j)\n", " g1 += Fxy(y^(j-m)*f*G)\n", " return(g1)\n", " \n", "class superelliptic_function:\n", " def __init__(self, C, g):\n", " p = C.characteristic\n", " Rxy. = PolynomialRing(GF(p), 2)\n", " Fxy = FractionField(Rxy)\n", " f = C.polynomial\n", " r = f.degree()\n", " m = C.exponent\n", " \n", " self.curve = C\n", " g = reduction(C, g)\n", " self.function = g\n", " \n", " def __repr__(self):\n", " return str(self.function)\n", " \n", " def jth_component(self, j):\n", " g = self.function\n", " C = self.curve\n", " p = C.characteristic\n", " Rx. = PolynomialRing(GF(p))\n", " Fx. = FractionField(Rx)\n", " FxRy. = PolynomialRing(Fx)\n", " g = FxRy(g)\n", " return coff(g, j)\n", " \n", " def __add__(self, other):\n", " C = self.curve\n", " g1 = self.function\n", " g2 = other.function\n", " g = reduction(C, g1 + g2)\n", " return superelliptic_function(C, g)\n", " \n", " def __sub__(self, other):\n", " C = self.curve\n", " g1 = self.function\n", " g2 = other.function\n", " g = reduction(C, g1 - g2)\n", " return superelliptic_function(C, g)\n", " \n", " def __mul__(self, other):\n", " C = self.curve\n", " g1 = self.function\n", " g2 = other.function\n", " g = reduction(C, g1 * g2)\n", " return superelliptic_function(C, g)\n", " \n", " def __truediv__(self, other):\n", " C = self.curve\n", " g1 = self.function\n", " g2 = other.function\n", " g = reduction(C, g1 / g2)\n", " return superelliptic_function(C, g)\n", " \n", "def diffn(self):\n", " C = self.curve\n", " f = C.polynomial\n", " m = C.exponent\n", " p = C.characteristic\n", " g = self.function\n", " Rxy. = PolynomialRing(GF(p), 2)\n", " Fxy = FractionField(Rxy)\n", " g = Fxy(g)\n", " A = g.derivative(x)\n", " B = g.derivative(y)*f.derivative(x)/(m*y^(m-1))\n", " return superelliptic_form(C, A+B)\n", " \n", "class superelliptic_form:\n", " def __init__(self, C, g):\n", " p = C.characteristic\n", " Rxy. = PolynomialRing(GF(p), 2)\n", " Fxy = FractionField(Rxy)\n", " g = Fxy(reduction_form(C, g))\n", " self.form = g\n", " self.curve = C \n", " \n", " def __add__(self, other):\n", " C = self.curve\n", " g1 = self.form\n", " g2 = other.form\n", " g = reduction(C, g1 + g2)\n", " return superelliptic_form(C, g)\n", " \n", " def __sub__(self, other):\n", " C = self.curve\n", " g1 = self.form\n", " g2 = other.form\n", " g = reduction(C, g1 - g2)\n", " return superelliptic_form(C, g)\n", " \n", " def __repr__(self):\n", " g = self.form\n", " if len(str(g)) == 1:\n", " return str(g) + ' dx'\n", " return '('+str(g) + ') dx'\n", "\n", " def __rmul__(self, constant):\n", " C = self.curve\n", " omega = self.form\n", " return superelliptic_form(C, constant*omega) \n", " \n", " def cartier(self):\n", " C = self.curve\n", " m = C.exponent\n", " p = C.characteristic\n", " f = C.polynomial\n", " Rx. = PolynomialRing(GF(p))\n", " Fx = FractionField(Rx)\n", " FxRy. = PolynomialRing(Fx)\n", " Fxy = FractionField(FxRy)\n", " result = superelliptic_form(C, FxRy(0))\n", " mult_order = Integers(m)(p).multiplicative_order()\n", " M = Integer((p^(mult_order)-1)/m)\n", " \n", " for j in range(1, m):\n", " fct_j = self.jth_component(j)\n", " h = Rx(fct_j*f^(M*j))\n", " j1 = (p^(mult_order-1)*j)%m\n", " B = floor(p^(mult_order-1)*j/m)\n", " result += superelliptic_form(C, polynomial_part(p, h)/(f^B*y^(j1)))\n", " return result \n", " \n", " def coordinates(self):\n", " C = self.curve\n", " p = C.characteristic\n", " m = C.exponent\n", " Rx. = PolynomialRing(GF(p))\n", " Fx = FractionField(Rx)\n", " FxRy. = PolynomialRing(Fx)\n", " g = C.genus()\n", " degrees_holo = C.degree_holo\n", " degrees_holo_inv = {b:a for a, b in degrees_holo.items()}\n", " basis = C.basis_holomorphic_differentials\n", " \n", " for j in range(1, m):\n", " omega_j = Fx(self.jth_component(j))\n", " if omega_j != Fx(0):\n", " d = degree_of_rational_fctn(omega_j, p)\n", " index = degrees_holo_inv[(d, j)]\n", " a = coeff_of_rational_fctn(omega_j, p)\n", " a1 = coeff_of_rational_fctn(basis[index].jth_component(j), p)\n", " elt = self - (a/a1)*basis[index]\n", " return elt.coordinates() + a/a1*vector([GF(p)(i == index) for i in range(0, g)])\n", " \n", " return vector(g*[0])\n", " \n", " def jth_component(self, j):\n", " g = self.form\n", " C = self.curve\n", " p = C.characteristic\n", " Rx. = PolynomialRing(GF(p))\n", " Fx = FractionField(Rx)\n", " FxRy. = PolynomialRing(Fx)\n", " Fxy = FractionField(FxRy)\n", " Ryinv. = PolynomialRing(Fx)\n", " g = Fxy(g)\n", " g = g(y = 1/y_inv)\n", " g = Ryinv(g)\n", " return coff(g, j)\n", " \n", " def is_regular_on_U0(self):\n", " C = self.curve\n", " p = C.characteristic\n", " m = C.exponent\n", " Rx. = PolynomialRing(GF(p))\n", " for j in range(1, m):\n", " if self.jth_component(j) not in Rx:\n", " return 0\n", " return 1\n", " \n", " def is_regular_on_Uinfty(self):\n", " C = self.curve\n", " p = C.characteristic\n", " m = C.exponent\n", " f = C.polynomial\n", " r = f.degree()\n", " delta = GCD(m, r)\n", " M = m/delta\n", " R = r/delta\n", " \n", " for j in range(1, m):\n", " A = self.jth_component(j)\n", " d = degree_of_rational_fctn(A, p)\n", " if(-d*M + j*R -(M+1)<0):\n", " return 0\n", " return 1\n", " \n", " \n", "class superelliptic_cech:\n", " def __init__(self, C, omega, fct):\n", " self.omega0 = omega\n", " self.omega8 = omega - diffn(fct)\n", " self.f = fct\n", " self.curve = C\n", " \n", " def __add__(self, other):\n", " C = self.curve\n", " return superelliptic_cech(C, self.omega0 + other.omega0, self.f + other.f)\n", " \n", " def __sub__(self, other):\n", " C = self.curve\n", " return superelliptic_cech(C, self.omega0 - other.omega0, self.f - other.f)\n", "\n", " def __rmul__(self, constant):\n", " C = self.curve\n", " w1 = self.omega0.form\n", " f1 = self.f.function\n", " w2 = superelliptic_form(C, constant*w1)\n", " f2 = superelliptic_function(C, constant*f1)\n", " return superelliptic_cech(C, w2, f2) \n", " \n", " def __repr__(self):\n", " return \"(\" + str(self.omega0) + \", \" + str(self.f) + \", \" + str(self.omega8) + \")\" \n", " \n", " def verschiebung(self):\n", " C = self.curve\n", " omega = self.omega0\n", " p = C.characteristic\n", " Rx. = PolynomialRing(GF(p))\n", " return superelliptic_cech(C, omega.cartier(), superelliptic_function(C, Rx(0)))\n", " \n", " def frobenius(self):\n", " C = self.curve\n", " fct = self.f.function\n", " p = C.characteristic\n", " Rx. = PolynomialRing(GF(p))\n", " return superelliptic_cech(C, superelliptic_form(C, Rx(0)), superelliptic_function(C, fct^p))\n", "\n", " def coordinates(self):\n", " C = self.curve\n", " p = C.characteristic\n", " m = C.exponent\n", " Rx. = PolynomialRing(GF(p))\n", " Fx = FractionField(Rx)\n", " FxRy. = PolynomialRing(Fx)\n", " g = C.genus()\n", " degrees_holo = C.degree_holo\n", " degrees_holo_inv = {b:a for a, b in degrees_holo.items()}\n", " degrees0 = C.degree_de_rham0\n", " degrees0_inv = {b:a for a, b in degrees0.items()}\n", " degrees1 = C.degree_de_rham1\n", " degrees1_inv = {b:a for a, b in degrees1.items()}\n", " basis = C.basis_de_rham\n", " \n", " omega = self.omega0\n", " fct = self.f\n", " \n", " if fct.function == Rx(0) and omega.form != Rx(0):\n", " for j in range(1, m):\n", " omega_j = Fx(omega.jth_component(j))\n", " if omega_j != Fx(0):\n", " d = degree_of_rational_fctn(omega_j, p)\n", " index = degrees_holo_inv[(d, j)]\n", " a = coeff_of_rational_fctn(omega_j, p)\n", " a1 = coeff_of_rational_fctn(basis[index].omega0.jth_component(j), p)\n", " elt = self - (a/a1)*basis[index]\n", " return elt.coordinates() + a/a1*vector([GF(p)(i == index) for i in range(0, 2*g)])\n", " \n", " for j in range(1, m):\n", " fct_j = Fx(fct.jth_component(j))\n", " if (fct_j != Rx(0)):\n", " d = degree_of_rational_fctn(fct_j, p)\n", " \n", " if (d, j) in degrees1.values():\n", " index = degrees1_inv[(d, j)]\n", " a = coeff_of_rational_fctn(fct_j, p)\n", " elt = self - (a/m)*basis[index]\n", " return elt.coordinates() + a/m*vector([GF(p)(i == index) for i in range(0, 2*g)])\n", " \n", " if d<0:\n", " a = coeff_of_rational_fctn(fct_j, p)\n", " h = superelliptic_function(C, FxRy(a*y^j*x^d))\n", " elt = superelliptic_cech(C, self.omega0, self.f - h)\n", " return elt.coordinates()\n", " \n", " if (fct_j != Rx(0)):\n", " G = superelliptic_function(C, y^j*x^d)\n", " a = coeff_of_rational_fctn(fct_j, p)\n", " elt =self - a*superelliptic_cech(C, diffn(G), G)\n", " return elt.coordinates()\n", "\n", " return vector(2*g*[0])\n", " \n", " def is_cocycle(self):\n", " w0 = self.omega0\n", " w8 = self.omega8\n", " fct = self.f\n", " if not w0.is_regular_on_U0() and not w8.is_regular_on_Uinfty():\n", " return('w0 & w8')\n", " if not w0.is_regular_on_U0():\n", " return('w0')\n", " if not w8.is_regular_on_Uinfty():\n", " return('w8')\n", " if w0.is_regular_on_U0() and w8.is_regular_on_Uinfty():\n", " return 1\n", " return 0\n", " \n", "def degree_of_rational_fctn(f, p):\n", " Rx. = PolynomialRing(GF(p))\n", " Fx = FractionField(Rx)\n", " f = Fx(f)\n", " f1 = f.numerator()\n", " f2 = f.denominator()\n", " d1 = f1.degree()\n", " d2 = f2.degree()\n", " return(d1 - d2)\n", "\n", "def coeff_of_rational_fctn(f, p):\n", " Rx. = PolynomialRing(GF(p))\n", " Fx = FractionField(Rx)\n", " f = Fx(f)\n", " if f == Rx(0):\n", " return 0\n", " f1 = f.numerator()\n", " f2 = f.denominator()\n", " d1 = f1.degree()\n", " d2 = f2.degree()\n", " a1 = f1.coefficients(sparse = false)[d1]\n", " a2 = f2.coefficients(sparse = false)[d2]\n", " return(a1/a2)\n", "\n", "def coff(f, d):\n", " lista = f.coefficients(sparse = false)\n", " if len(lista) <= d:\n", " return 0\n", " return lista[d]\n", "\n", "def cut(f, i):\n", " R = f.parent()\n", " coeff = f.coefficients(sparse = false)\n", " return sum(R(x^(j-i-1)) * coeff[j] for j in range(i+1, f.degree() + 1))\n", "\n", "def polynomial_part(p, h):\n", " Rx. = PolynomialRing(GF(p))\n", " h = Rx(h)\n", " result = Rx(0)\n", " for i in range(0, h.degree()+1):\n", " if (i%p) == p-1:\n", " power = Integer((i-(p-1))/p)\n", " result += Integer(h[i]) * x^(power) \n", " return result" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "def preimage(U, V, M): #preimage of subspace U under M\n", " basis_preimage = M.right_kernel().basis()\n", " imageU = U.intersection(M.transpose().image())\n", " basis = imageU.basis()\n", " for v in basis:\n", " w = M.solve_right(v)\n", " basis_preimage = basis_preimage + [w]\n", " return V.subspace(basis_preimage)\n", "\n", "def image(U, V, M):\n", " basis = U.basis()\n", " basis_image = []\n", " for v in basis:\n", " basis_image += [M*v]\n", " return V.subspace(basis_image)\n", "\n", "def flag(F, V, p, test = 0):\n", " dim = F.dimensions()[0]\n", " space = VectorSpace(GF(p), dim)\n", " flag_subspaces = (dim+1)*[0]\n", " flag_used = (dim+1)*[0]\n", " final_type = (dim+1)*['?']\n", " \n", " flag_subspaces[dim] = space\n", " flag_used[dim] = 1\n", " \n", " \n", " while 1 in flag_used:\n", " index = flag_used.index(1)\n", " flag_used[index] = 0\n", " U = flag_subspaces[index]\n", " U_im = image(U, space, V)\n", " d_im = U_im.dimension()\n", " final_type[index] = d_im\n", " U_pre = preimage(U, space, F)\n", " d_pre = U_pre.dimension()\n", " \n", " if flag_subspaces[d_im] == 0:\n", " flag_subspaces[d_im] = U_im\n", " flag_used[d_im] = 1\n", " \n", " if flag_subspaces[d_pre] == 0:\n", " flag_subspaces[d_pre] = U_pre\n", " flag_used[d_pre] = 1\n", " \n", " if test == 1:\n", " print('(', final_type, ')')\n", " \n", " for i in range(0, dim+1):\n", " if final_type[i] == '?' and final_type[dim - i] != '?':\n", " i1 = dim - i\n", " final_type[i] = final_type[i1] - i1 + dim/2\n", " \n", " final_type[0] = 0\n", " \n", " for i in range(1, dim+1):\n", " if final_type[i] == '?':\n", " prev = final_type[i-1]\n", " if prev != '?' and prev in final_type[i+1:]:\n", " final_type[i] = prev\n", " \n", " for i in range(1, dim+1):\n", " if final_type[i] == '?':\n", " final_type[i] = min(final_type[i-1] + 1, dim/2)\n", " \n", " if is_final(final_type, dim/2):\n", " return final_type[1:dim/2 + 1]\n", " print('error:', final_type[1:dim/2 + 1])\n", " \n", "def is_final(final_type, dim):\n", " n = len(final_type)\n", " if final_type[0] != 0:\n", " return 0\n", " \n", " if final_type[n-1] != dim:\n", " return 0\n", " \n", " for i in range(1, n):\n", " if final_type[i] != final_type[i - 1] and final_type[i] != final_type[i - 1] + 1:\n", " return 0\n", " return 1" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "def dzialanie(f, m, p):\n", " Rx. = PolynomialRing(GF(p))\n", " fp = Rx(f(x^p))\n", " C = superelliptic(fp, m, p)\n", " holo = C.basis_holomorphic_differentials\n", " \n", " Rxy. = PolynomialRing(GF(p), 2)\n", " kxi1. = PolynomialRing(FractionField(Rxy))\n", " kxi = kxi1.quotient(xi1^p)\n", " xi = kxi(xi1)\n", " holo_forms = [a.form for a in holo]\n", " holo_xi = [kxi(a(x = x+xi, y = y)) for a in holo_forms]\n", " \n", " N = matrix(kxi1, C.genus(), C.genus())\n", " for i in range(0, p):\n", " M = matrix(GF(p), C.genus(), C.genus())\n", " for j in range(0, len(holo_xi)):\n", " a = holo_xi[j]\n", " omega = superelliptic_form(C, a[i])\n", " v = omega.coordinates()\n", " M[j, :] = v\n", " N += xi1^i*M\n", " return N\n", "\n", "def bloki(A):\n", " B = A.jordan_form(subdivide=True)\n", " lista = []\n", " d = 0\n", " i = 0\n", " while d < B.dimensions()[1]:\n", " lista.append(B.subdivision(i, i).dimensions()[1])\n", " d = d + B.subdivision(i, i).dimensions()[1]\n", " i = i+1\n", " return lista\n", "\n", "def p_cov(C):\n", " m = C.exponent\n", " p = C.characteristic\n", " f = C.polynomial\n", " return superelliptic(f(x^p), m, p)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "# Testy" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4, 3, 3, 3, 2, 2, 2, 1, 1, 1]\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "10 76\n" ] } ], "source": [ "p = 5\n", "Rx. = PolynomialRing(GF(p))\n", "r = 3\n", "f = x^(r) + x + 1\n", "m = 12\n", "A = dzialanie(f, m, p)\n", "print(bloki(A))\n", "C = superelliptic(f, m, p)\n", "print(C.genus(), p_cov(C).genus())" ] }, { "cell_type": "code", "execution_count": 79, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "omega = C.basis_holomorphic_differentials[2]" ] }, { "cell_type": "code", "execution_count": 81, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "lista = [a.form for a in C.basis_holomorphic_differentials]" ] }, { "cell_type": "code", "execution_count": 82, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "kxi1. = PolynomialRing(FractionField(Rxy))\n", "kxi = kxi1.quotient(xi1^p)\n", "xi = kxi(xi1)" ] }, { "cell_type": "code", "execution_count": 96, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "lista2 = [kxi(a(x = x+xi, y = y)) for a in lista]" ] }, { "cell_type": "code", "execution_count": 92, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "x" ] }, "execution_count": 92, "metadata": { }, "output_type": "execute_result" } ], "source": [ "kxi(x)" ] }, { "cell_type": "code", "execution_count": 98, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "a = lista2[0]" ] }, { "cell_type": "code", "execution_count": 99, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "1/y*xi1bar^2 + 2*x/y*xi1bar + x^2/y" ] }, "execution_count": 99, "metadata": { }, "output_type": "execute_result" } ], "source": [ "omega.form(x = x+xi, y = y)" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "ename": "TypeError", "evalue": "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", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mAttributeError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/rings/fraction_field.py\u001b[0m in \u001b[0;36m_element_constructor_\u001b[0;34m(self, x, y, coerce)\u001b[0m\n\u001b[1;32m 695\u001b[0m \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 696\u001b[0;31m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0my\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mresolve_fractions\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx0\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0my0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 697\u001b[0m \u001b[0;32mexcept\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mAttributeError\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/rings/fraction_field.py\u001b[0m in \u001b[0;36mresolve_fractions\u001b[0;34m(x, y)\u001b[0m\n\u001b[1;32m 672\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mresolve_fractions\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0my\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 673\u001b[0;31m \u001b[0mxn\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mnumerator\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 674\u001b[0m \u001b[0mxd\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdenominator\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/structure/element.pyx\u001b[0m in \u001b[0;36msage.structure.element.Element.__getattr__ (build/cythonized/sage/structure/element.c:4754)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 493\u001b[0m \"\"\"\n\u001b[0;32m--> 494\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mgetattr_from_category\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mname\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 495\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/structure/element.pyx\u001b[0m in \u001b[0;36msage.structure.element.Element.getattr_from_category (build/cythonized/sage/structure/element.c:4866)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 506\u001b[0m \u001b[0mcls\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mP\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_abstract_element_class\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 507\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mgetattr_from_other_class\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mname\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 508\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/cpython/getattr.pyx\u001b[0m in \u001b[0;36msage.cpython.getattr.getattr_from_other_class (build/cythonized/sage/cpython/getattr.c:2566)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 355\u001b[0m \u001b[0mdummy_error_message\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mname\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mname\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 356\u001b[0;31m \u001b[0;32mraise\u001b[0m \u001b[0mAttributeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdummy_error_message\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 357\u001b[0m \u001b[0mcdef\u001b[0m \u001b[0mPyObject\u001b[0m\u001b[0;34m*\u001b[0m \u001b[0mattr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0minstance_getattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mname\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;31mAttributeError\u001b[0m: 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomialRing_libsingular' object has no attribute '__custom_name'", "\nDuring handling of the above exception, another exception occurred:\n", "\u001b[0;31mTypeError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m/tmp/ipykernel_1899/916120484.py\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0momega1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msuperelliptic_form\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0momega\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mform\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0mxi\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0my\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0my\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m", "\u001b[0;32m/tmp/ipykernel_1899/3188561669.py\u001b[0m in \u001b[0;36m__init__\u001b[0;34m(self, C, g)\u001b[0m\n\u001b[1;32m 283\u001b[0m \u001b[0mRxy\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mPolynomialRing\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mGF\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnames\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'x'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'y'\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m;\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0my\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mRxy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_first_ngens\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 284\u001b[0m \u001b[0mFxy\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mFractionField\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mRxy\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 285\u001b[0;31m \u001b[0mg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mFxy\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mreduction_form\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 286\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mform\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 287\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcurve\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;32m/tmp/ipykernel_1899/3188561669.py\u001b[0m in \u001b[0;36mreduction_form\u001b[0;34m(C, g)\u001b[0m\n\u001b[1;32m 194\u001b[0m \u001b[0mr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdegree\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 195\u001b[0m \u001b[0mm\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexponent\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 196\u001b[0;31m \u001b[0mg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mreduction\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 197\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 198\u001b[0m \u001b[0mg1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mRxy\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;32m/tmp/ipykernel_1899/3188561669.py\u001b[0m in \u001b[0;36mreduction\u001b[0;34m(C, g)\u001b[0m\n\u001b[1;32m 169\u001b[0m \u001b[0mr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdegree\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 170\u001b[0m \u001b[0mm\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexponent\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 171\u001b[0;31m \u001b[0mg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mFxy\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 172\u001b[0m \u001b[0mg1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mnumerator\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 173\u001b[0m \u001b[0mg2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdenominator\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/structure/parent.pyx\u001b[0m in \u001b[0;36msage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9388)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 896\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mmor\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 897\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mno_extra_args\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 898\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mmor\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_call_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 899\u001b[0m \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 900\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mmor\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_call_with_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/structure/coerce_maps.pyx\u001b[0m in \u001b[0;36msage.structure.coerce_maps.DefaultConvertMap_unique._call_ (build/cythonized/sage/structure/coerce_maps.c:4665)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 159\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 160\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_element_constructor\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_element_constructor\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 161\u001b[0;31m \u001b[0;32mraise\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 162\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 163\u001b[0m \u001b[0mcpdef\u001b[0m \u001b[0mElement\u001b[0m \u001b[0m_call_with_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mkwds\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m{\u001b[0m\u001b[0;34m}\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/structure/coerce_maps.pyx\u001b[0m in \u001b[0;36msage.structure.coerce_maps.DefaultConvertMap_unique._call_ (build/cythonized/sage/structure/coerce_maps.c:4557)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 154\u001b[0m \u001b[0mcdef\u001b[0m \u001b[0mParent\u001b[0m \u001b[0mC\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_codomain\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 155\u001b[0m \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 156\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_element_constructor\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 157\u001b[0m \u001b[0;32mexcept\u001b[0m \u001b[0mException\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 158\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mprint_warnings\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/rings/fraction_field.py\u001b[0m in \u001b[0;36m_element_constructor_\u001b[0;34m(self, x, y, coerce)\u001b[0m\n\u001b[1;32m 696\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0my\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mresolve_fractions\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx0\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0my0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 697\u001b[0m \u001b[0;32mexcept\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mAttributeError\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 698\u001b[0;31m raise TypeError(\"cannot convert {!r}/{!r} to an element of {}\".format(\n\u001b[0m\u001b[1;32m 699\u001b[0m x0, y0, self))\n\u001b[1;32m 700\u001b[0m \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;31mTypeError\u001b[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" ] } ], "source": [ "omega1 = superelliptic_form(C, omega.form(x = x+xi, y = y))" ] }, { "cell_type": "code", "execution_count": 68, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "kxi1. = PolynomialRing(GF(p))\n", "kxi = FractionField(kxi1)\n", "Rxy. = PolynomialRing(kxi, 2)\n", "Fxy = FractionField(Rxy)\n", "\n", "Rx. = PolynomialRing(kxi)\n", "Fx = FractionField(Rx)\n", "FxRy. = PolynomialRing(Fx)" ] }, { "cell_type": "code", "execution_count": 69, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "x/y" ] }, "execution_count": 69, "metadata": { }, "output_type": "execute_result" } ], "source": [ "C.basis_holomorphic_differentials" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]\n", "[ xi1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]\n", "[ xi1^2 2*xi1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0]\n", "[ xi1^3 -2*xi1^2 -2*xi1 1 0 0 0 0 0 0 0 0 0 0 0 0 0]\n", "[ xi1^4 -xi1^3 xi1^2 -xi1 1 0 0 0 0 0 0 0 0 0 0 0 0]\n", "[ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0]\n", "[ 0 0 0 0 0 xi1 1 0 0 0 0 0 0 0 0 0 0]\n", "[ 0 0 0 0 0 xi1^2 2*xi1 1 0 0 0 0 0 0 0 0 0]\n", "[ 0 0 0 0 0 xi1^3 -2*xi1^2 -2*xi1 1 0 0 0 0 0 0 0 0]\n", "[ 0 0 0 0 0 xi1^4 -xi1^3 xi1^2 -xi1 1 0 0 0 0 0 0 0]\n", "[ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0]\n", "[ 0 0 0 0 0 0 0 0 0 0 xi1 1 0 0 0 0 0]\n", "[ 0 0 0 0 0 0 0 0 0 0 xi1^2 2*xi1 1 0 0 0 0]\n", "[ 0 0 0 0 0 0 0 0 0 0 xi1^3 -2*xi1^2 -2*xi1 1 0 0 0]\n", "[ 0 0 0 0 0 0 0 0 0 0 xi1^4 -xi1^3 xi1^2 -xi1 1 0 0]\n", "[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0]\n", "[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 xi1 1]\n" ] } ], "source": [ "p = 5\n", "Rx. = PolynomialRing(GF(p))\n", "f = Rx(x^7 + x + 1)\n", "m = 2\n", "Rxy. = PolynomialRing(GF(p), 2)\n", "print(dzialanie(f, m, p))\n", "A = dzialanie(f, m, p)" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[1 1 0 0 0|0 0 0 0 0|0 0 0 0 0|0 0]\n", "[0 1 1 0 0|0 0 0 0 0|0 0 0 0 0|0 0]\n", "[0 0 1 1 0|0 0 0 0 0|0 0 0 0 0|0 0]\n", "[0 0 0 1 1|0 0 0 0 0|0 0 0 0 0|0 0]\n", "[0 0 0 0 1|0 0 0 0 0|0 0 0 0 0|0 0]\n", "[---------+---------+---------+---]\n", "[0 0 0 0 0|1 1 0 0 0|0 0 0 0 0|0 0]\n", "[0 0 0 0 0|0 1 1 0 0|0 0 0 0 0|0 0]\n", "[0 0 0 0 0|0 0 1 1 0|0 0 0 0 0|0 0]\n", "[0 0 0 0 0|0 0 0 1 1|0 0 0 0 0|0 0]\n", "[0 0 0 0 0|0 0 0 0 1|0 0 0 0 0|0 0]\n", "[---------+---------+---------+---]\n", "[0 0 0 0 0|0 0 0 0 0|1 1 0 0 0|0 0]\n", "[0 0 0 0 0|0 0 0 0 0|0 1 1 0 0|0 0]\n", "[0 0 0 0 0|0 0 0 0 0|0 0 1 1 0|0 0]\n", "[0 0 0 0 0|0 0 0 0 0|0 0 0 1 1|0 0]\n", "[0 0 0 0 0|0 0 0 0 0|0 0 0 0 1|0 0]\n", "[---------+---------+---------+---]\n", "[0 0 0 0 0|0 0 0 0 0|0 0 0 0 0|1 1]\n", "[0 0 0 0 0|0 0 0 0 0|0 0 0 0 0|0 1]" ] }, "execution_count": 11, "metadata": { }, "output_type": "execute_result" } ], "source": [ "A.jordan_form()" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "M = matrix(Rx, 3,3)" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "M = matrix(Rx, [[x, 1, 1], [1,2,3], [x+1,2,4]])" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "3" ] }, "execution_count": 25, "metadata": { }, "output_type": "execute_result" } ], "source": [ "M.rank()" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "kxi1. = PolynomialRing(FractionField(Rxy))\n", "kxi = kxi1.quotient(xi1^p)\n", "xi = kxi(xi1)" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0" ] }, "execution_count": 33, "metadata": { }, "output_type": "execute_result" } ], "source": [ "lift(xi^5)" ] }, { "cell_type": "code", "execution_count": 35, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "E = EllipticCurve(GF(3), [1,1])" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.5", "language": "sagemath", "metadata": { "cocalc": { "description": "Open-source mathematical software system", "priority": 10, "url": "https://www.sagemath.org/" } }, "name": "sage-9.5", "resource_dir": "/ext/jupyter/kernels/sage-9.5" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.9.9" } }, "nbformat": 4, "nbformat_minor": 4 }