macierz froba prawie dziala; problem z najstarszym wspol
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@ -22,7 +22,7 @@
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"cell_type": "code",
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"execution_count": 169,
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"execution_count": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -46,7 +46,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 170,
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"execution_count": 3,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -79,7 +79,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 208,
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"execution_count": 4,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -213,7 +213,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 209,
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"execution_count": 5,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -261,7 +261,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 242,
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"execution_count": 6,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -456,6 +456,30 @@
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"baza_dr(m, f, 0, p)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"{0: [1, 0, 3], 1: [0, 2/x, 3]}"
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]
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},
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"execution_count": 8,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"p = 5\n",
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"R.<x> = PolynomialRing(GF(p))\n",
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"f = x^3 + x + 2\n",
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"m = 7\n",
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"baza_dr(m, f, 3, p)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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@ -2,7 +2,7 @@
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 115,
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"execution_count": 118,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -75,7 +75,7 @@
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" if (r*(m-j) - m*i >= delta):\n",
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" s = Rx(m-j)*Rx(x)*Rx(f.derivative()) - Rx(m)*Rx(i)*f\n",
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" psi = Rx(cut(s, i))\n",
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" basis[t] = superelliptic_cech(self, superelliptic_form(self, Fxy(psi/y^j)), superelliptic_function(self, Fxy(m*y^j/x^i)))\n",
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" basis[t] = superelliptic_cech(self, superelliptic_form(self, Fxy(psi/y^j)), superelliptic_function(self, Fxy(m*y^(m-j)/x^i)))\n",
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" degrees0[t] = (psi.degree(), j)\n",
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" degrees1[t] = (-i, j)\n",
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" t += 1\n",
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@ -92,6 +92,27 @@
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" def basis_de_rham(self, j = 'all'): \n",
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" basis, degrees0, degrees1 = self.degree_and_basis_de_rham(j)\n",
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" return basis\n",
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" \n",
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" def verschiebung_matrix(self):\n",
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" basis = self.basis_de_rham()\n",
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" g = self.genus()\n",
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" p = self.characteristic\n",
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" M = matrix(GF(p), 2*g, 2*g)\n",
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" for i, w in basis.items():\n",
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" v = w.verschiebung().coordinates()\n",
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" M[i, :] = v\n",
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" return M\n",
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" \n",
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" def frobenius_matrix(self):\n",
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" basis = self.basis_de_rham()\n",
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" g = self.genus()\n",
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" p = self.characteristic\n",
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" M = matrix(GF(p), 2*g, 2*g)\n",
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" for i, w in basis.items():\n",
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" print('w', w)\n",
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" v = w.frobenius().coordinates()\n",
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" M[i, :] = v\n",
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" return M\n",
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" \n",
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"def reduction(C, g):\n",
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" p = C.characteristic\n",
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@ -238,6 +259,27 @@
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" return str(g) + ' dx'\n",
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" return '('+str(g) + ') dx'\n",
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" \n",
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" def cartier(self):\n",
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" C = self.curve\n",
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" m = C.exponent\n",
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" p = C.characteristic\n",
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" f = C.polynomial\n",
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" Rx.<x> = PolynomialRing(GF(p))\n",
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" Fx = FractionField(Rx)\n",
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" FxRy.<y> = PolynomialRing(Fx)\n",
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" Fxy = FractionField(FxRy)\n",
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" result = superelliptic_form(C, FxRy(0))\n",
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" mult_order = Integers(m)(p).multiplicative_order()\n",
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" M = Integer((p^(mult_order)-1)/m)\n",
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" \n",
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" for j in range(1, m):\n",
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" fct_j = self.jth_component(j)\n",
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" h = Rx(fct_j*f^(M*j))\n",
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" j1 = (p^(mult_order-1)*j)%m\n",
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" B = floor(p^(mult_order-1)*j/m)\n",
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" result += superelliptic_form(C, polynomial_part(p, h)/(f^B*y^(j1)))\n",
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" return result\n",
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" \n",
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" def jth_component(self, j):\n",
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" g = self.form\n",
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" C = self.curve\n",
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@ -276,6 +318,7 @@
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" A = self.jth_component(j)\n",
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" d = degree_of_rational_fctn(A)\n",
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" if(-d*M + j*R -(M+1)<0):\n",
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" print('not', d, j, r, m, -d*M + j*R -(M+1))\n",
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" return 0\n",
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" return 1\n",
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" \n",
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@ -306,13 +349,22 @@
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" def __repr__(self):\n",
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" return \"(\" + str(self.omega0) + \", \" + str(self.f) + \", \" + str(self.omega8) + \")\" \n",
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" \n",
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" def basis_coeffs(self):\n",
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" def verschiebung(self):\n",
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" C = self.curve\n",
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" g = self.f\n",
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" basis = C.basis_de_rham()\n",
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" omega = self.omega0\n",
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" p = C.characteristic\n",
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" Rx.<x> = PolynomialRing(GF(p))\n",
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" return superelliptic_cech(C, omega.cartier(), superelliptic_function(C, Rx(0)))\n",
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" \n",
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" def frobenius(self):\n",
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" C = self.curve\n",
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" fct = self.f.function\n",
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" p = C.characteristic\n",
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" Rx.<x> = PolynomialRing(GF(p))\n",
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" return superelliptic_cech(C, superelliptic_form(C, Rx(0)), superelliptic_function(C, fct^p))\n",
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"\n",
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" def coordinates(self):\n",
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" print(self)\n",
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" print(self, self.is_cocycle())\n",
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" C = self.curve\n",
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" p = C.characteristic\n",
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" m = C.exponent\n",
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@ -329,41 +381,63 @@
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" omega = self.omega0\n",
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" fct = self.f\n",
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" \n",
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" if fct.function == Rx(0) and omega.form == Rx(0):\n",
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" if fct.function == Rx(0) and omega.form != Rx(0):\n",
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" print('c1')\n",
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" for j in range(1, m):\n",
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" omega_j = Fx(omega.jth_component(j))\n",
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" if omega_j != Fx(0):\n",
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" d = d = degree_of_rational_fctn(omega_j)\n",
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" d = degree_of_rational_fctn(omega_j)\n",
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" index = degrees0_inv[(d, j)]\n",
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" print('baza', basis[index])\n",
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" a = coeff_of_rational_fctn(omega_j)\n",
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" a1 = coeff_of_rational_fctn(basis[index].omega0.jth_component(j))\n",
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" elt = self - basis[index].mult(a/a1)\n",
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" return elt.coordinates() + a/a1*vector([GF(p)(i == index) for i in range(0, 2*g)])\n",
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" \n",
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" for j in range(1, m):\n",
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" print('c2')\n",
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" fct_j = Fx(fct.jth_component(j))\n",
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" if (fct_j != Rx(0)):\n",
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" d = degree_of_rational_fctn(fct_j)\n",
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" \n",
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" if (d, j) in degrees1.values():\n",
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" print('c2a')\n",
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" index = degrees1_inv[(d, j)]\n",
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" a = coeff_of_rational_fctn(fct_j)\n",
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" a1 = coeff_of_rational_fctn(basis[index].f.jth_component(j))\n",
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" print(a, a1, index, basis[index].is_cocycle())\n",
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" elt = self - basis[index].mult(a/a1)\n",
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" return elt.coordinates() + a/a1*vector([GF(p)(i == index) for i in range(0, 2*g)])\n",
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" \n",
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" if d<0:\n",
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" print('c2b')\n",
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" a = coeff_of_rational_fctn(fct_j)\n",
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" elt =- superelliptic_cech(C, elt.omega0, elt.f - FxRy(y^j*x^d)).mult(a)\n",
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" h = superelliptic_function(C, FxRy(a*y^j*x^d))\n",
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" elt = superelliptic_cech(C, self.omega0, self.f - h)\n",
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" return elt.coordinates()\n",
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" \n",
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" if (fct_j != Rx(0)):\n",
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" print('c2c')\n",
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" G = superelliptic_function(C, y^j*x^d)\n",
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" a = coeff_of_rational_fctn(fct_j)\n",
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" elt =-superelliptic_cech(diffn(G), G).mult(a)\n",
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" elt =self - superelliptic_cech(C, diffn(G), G).mult(a)\n",
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" return elt.coordinates()\n",
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"\n",
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" return vector(2*g*[0])\n",
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" \n",
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" def is_cocycle(self):\n",
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" w0 = self.omega0\n",
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" w8 = self.omega8\n",
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" fct = self.f\n",
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" if not w0.is_regular_on_U0() and not w8.is_regular_on_Uinfty():\n",
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" return('w0 & w8')\n",
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" if not w0.is_regular_on_U0():\n",
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" return('w0')\n",
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" if not w8.is_regular_on_Uinfty():\n",
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" return('w8')\n",
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" if w0.is_regular_on_U0() and w8.is_regular_on_Uinfty():\n",
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" return 1\n",
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" return 0\n",
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" \n",
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"def degree_of_rational_fctn(f):\n",
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" Rx.<x> = PolynomialRing(GF(p))\n",
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@ -376,9 +450,12 @@
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" return(d1 - d2)\n",
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"\n",
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"def coeff_of_rational_fctn(f):\n",
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" print('coeff', f)\n",
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" Rx.<x> = PolynomialRing(GF(p))\n",
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" Fx = FractionField(Rx)\n",
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" f = Fx(f)\n",
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" if f == Rx(0):\n",
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" return 0\n",
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" f1 = f.numerator()\n",
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" f2 = f.denominator()\n",
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" d1 = f1.degree()\n",
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"def cut(f, i):\n",
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" R = f.parent()\n",
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" coeff = f.coefficients(sparse = false)\n",
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" return sum(R(x^(j-i-1)) * coeff[j] for j in range(i+1, f.degree() + 1))"
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" return sum(R(x^(j-i-1)) * coeff[j] for j in range(i+1, f.degree() + 1))\n",
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"\n",
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"def polynomial_part(p, h):\n",
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" Rx.<x> = PolynomialRing(GF(p))\n",
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" h = Rx(h)\n",
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" result = Rx(0)\n",
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" for i in range(0, h.degree()+1):\n",
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" if (i%p) == p-1:\n",
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" power = Integer((i-(p-1))/p)\n",
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" result += Integer(h[i]) * x^(power) \n",
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" return result"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 116,
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"execution_count": 119,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"{0: ((x/y) dx, 2/x*y, ((x^3*y^5 - x^3 + x - 1)/(x^2*y^6)) dx),\n",
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" 1: (((-1)/y) dx, 2/x^2*y, ((-x^3*y^5 + x^3 - 2*x - 2)/(x^3*y^6)) dx),\n",
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" 2: (((-2*x)/y^2) dx, 2/x*y^2, ((-2*x^3*y^3 + x^3 - 1)/(x^2*y^5)) dx),\n",
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" 3: ((1/y^2) dx, 2/x^2*y^2, ((x^3*y^3 - 2*x^3 + 2*x - 2)/(x^3*y^5)) dx),\n",
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" 4: ((1/y^3) dx, 0, (1/y^3) dx),\n",
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" 5: (0 dx, 2/x*y^3, ((-2*x^3 - x - 1)/(x^2*y^4)) dx),\n",
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" 6: ((1/y^4) dx, 0, (1/y^4) dx),\n",
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" 7: ((2*x/y^4) dx, 2/x*y^4, ((2*x^3 - 2*x*y - y)/(x^2*y^4)) dx),\n",
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" 8: ((1/y^5) dx, 0, (1/y^5) dx),\n",
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" 9: ((x/y^5) dx, 0, (x/y^5) dx),\n",
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" 10: ((1/y^6) dx, 0, (1/y^6) dx),\n",
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" 11: ((x/y^6) dx, 0, (x/y^6) dx)}"
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]
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},
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"execution_count": 116,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"outputs": [],
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"source": [
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"p = 5\n",
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"C = superelliptic(x^3 + x + 2, 7, p)\n",
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"C.basis_de_rham()\n",
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"baza = C.basis_de_rham()\n",
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"#C.basis_holomorphic_differentials()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 117,
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"execution_count": 120,
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"metadata": {},
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"outputs": [],
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"((x/y) dx, 2/x*y^6, ((x - 1)/(x^2*y)) dx) 1\n",
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"(((-1)/y) dx, 2/x^2*y^6, ((-2*x - 2)/(x^3*y)) dx) 1\n",
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"(((-2*x)/y^2) dx, 2/x*y^5, ((2*x - 1)/(x^2*y^2)) dx) 1\n",
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"((1/y^2) dx, 2/x^2*y^5, ((-x - 2)/(x^3*y^2)) dx) 1\n",
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"((1/y^3) dx, 0, (1/y^3) dx) 1\n",
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"(0 dx, 2/x*y^4, ((-2*x - 1)/(x^2*y^3)) dx) 1\n",
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"((1/y^4) dx, 0, (1/y^4) dx) 1\n",
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"((2*x/y^4) dx, 2/x*y^3, ((-x - 1)/(x^2*y^4)) dx) 1\n",
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"((1/y^5) dx, 0, (1/y^5) dx) 1\n",
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"((x/y^5) dx, 0, (x/y^5) dx) 1\n",
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"((1/y^6) dx, 0, (1/y^6) dx) 1\n",
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"((x/y^6) dx, 0, (x/y^6) dx) 1\n"
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]
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}
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],
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"source": [
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"RxRy.<x, y> = PolynomialRing(GF(p), 2)\n",
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"w1 = superelliptic_cech(C, superelliptic_form(C, (1/y^5)), superelliptic_function(C, 0))\n",
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"w2 = superelliptic_cech(C, superelliptic_form(C,2*x/y^4), superelliptic_function(C, 2/x*y^4))\n",
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"w = w1+w2+w2"
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"for w in baza.values():\n",
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" print(w, w.is_cocycle())"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 118,
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"execution_count": 75,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"({0: ((x/y) dx, 2/x*y, ((x^3*y^5 - x^3 + x - 1)/(x^2*y^6)) dx),\n",
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" 1: (((-1)/y) dx, 2/x^2*y, ((-x^3*y^5 + x^3 - 2*x - 2)/(x^3*y^6)) dx),\n",
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" 2: (((-2*x)/y^2) dx, 2/x*y^2, ((-2*x^3*y^3 + x^3 - 1)/(x^2*y^5)) dx),\n",
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" 3: ((1/y^2) dx, 2/x^2*y^2, ((x^3*y^3 - 2*x^3 + 2*x - 2)/(x^3*y^5)) dx),\n",
|
||||
" 4: ((1/y^3) dx, 0, (1/y^3) dx),\n",
|
||||
" 5: (0 dx, 2/x*y^3, ((-2*x^3 - x - 1)/(x^2*y^4)) dx),\n",
|
||||
" 6: ((1/y^4) dx, 0, (1/y^4) dx),\n",
|
||||
" 7: ((2*x/y^4) dx, 2/x*y^4, ((2*x^3 - 2*x*y - y)/(x^2*y^4)) dx),\n",
|
||||
" 8: ((1/y^5) dx, 0, (1/y^5) dx),\n",
|
||||
" 9: ((x/y^5) dx, 0, (x/y^5) dx),\n",
|
||||
" 10: ((1/y^6) dx, 0, (1/y^6) dx),\n",
|
||||
" 11: ((x/y^6) dx, 0, (x/y^6) dx)},\n",
|
||||
" {0: (1, 1),\n",
|
||||
" 1: (0, 1),\n",
|
||||
" 2: (1, 2),\n",
|
||||
" 3: (0, 2),\n",
|
||||
" 4: (0, 3),\n",
|
||||
" 5: (-1, 3),\n",
|
||||
" 6: (0, 4),\n",
|
||||
" 7: (1, 4),\n",
|
||||
" 8: (0, 5),\n",
|
||||
" 9: (1, 5),\n",
|
||||
" 10: (0, 6),\n",
|
||||
" 11: (1, 6)},\n",
|
||||
" {0: (-1, 1), 1: (-2, 1), 2: (-1, 2), 3: (-2, 2), 5: (-1, 3), 7: (-1, 4)})"
|
||||
]
|
||||
},
|
||||
"execution_count": 75,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"C.degree_and_basis_de_rham()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 121,
|
||||
"metadata": {
|
||||
"scrolled": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"w ((x/y) dx, 2/x*y^6, ((x - 1)/(x^2*y)) dx)\n",
|
||||
"(0 dx, ((2*x^12 + 3*x^10 + x^9 + 2*x^8 + 3*x^7 + x^6 + 3*x^5 + 3*x^4 + 3*x^2 + 4*x + 2)/x^5)*y^2, 0 dx) 1\n",
|
||||
"c2\n",
|
||||
"c2\n",
|
||||
"c2c\n",
|
||||
"coeff (2*x^12 + 3*x^10 + x^9 + 2*x^8 + 3*x^7 + x^6 + 3*x^5 + 3*x^4 + 3*x^2 + 4*x + 2)/x^5\n",
|
||||
"(((-x^7 + 2*x^6)/y^5) dx, ((3*x^10 + x^9 + 2*x^8 + 3*x^7 + x^6 + 3*x^5 + 3*x^4 + 3*x^2 + 4*x + 2)/x^5)*y^2, 0 dx) 1\n",
|
||||
"c2\n",
|
||||
"c2\n",
|
||||
"c2c\n",
|
||||
"coeff (3*x^10 + x^9 + 2*x^8 + 3*x^7 + x^6 + 3*x^5 + 3*x^4 + 3*x^2 + 4*x + 2)/x^5\n",
|
||||
"(((2*x^6 + 2*x^5)/y^5) dx, ((x^9 + 2*x^8 + 3*x^7 + x^6 + 3*x^5 + 3*x^4 + 3*x^2 + 4*x + 2)/x^5)*y^2, 0 dx) 1\n",
|
||||
"c2\n",
|
||||
"c2\n",
|
||||
"c2c\n",
|
||||
"coeff (x^9 + 2*x^8 + 3*x^7 + x^6 + 3*x^5 + 3*x^4 + 3*x^2 + 4*x + 2)/x^5\n",
|
||||
"(((2*x^5 + 2*x^3)/y^5) dx, ((2*x^8 + 3*x^7 + x^6 + 3*x^5 + 3*x^4 + 3*x^2 + 4*x + 2)/x^5)*y^2, 0 dx) 1\n",
|
||||
"c2\n",
|
||||
"c2\n",
|
||||
"c2c\n",
|
||||
"coeff (2*x^8 + 3*x^7 + x^6 + 3*x^5 + 3*x^4 + 3*x^2 + 4*x + 2)/x^5\n",
|
||||
"(((-x^3 - 2*x^2)/y^5) dx, ((3*x^7 + x^6 + 3*x^5 + 3*x^4 + 3*x^2 + 4*x + 2)/x^5)*y^2, 0 dx) 1\n",
|
||||
"c2\n",
|
||||
"c2\n",
|
||||
"c2c\n",
|
||||
"coeff (3*x^7 + x^6 + 3*x^5 + 3*x^4 + 3*x^2 + 4*x + 2)/x^5\n",
|
||||
"(((-x^3 - x^2 - 2*x)/y^5) dx, ((x^6 + 3*x^5 + 3*x^4 + 3*x^2 + 4*x + 2)/x^5)*y^2, 0 dx) 1\n",
|
||||
"c2\n",
|
||||
"c2\n",
|
||||
"c2c\n",
|
||||
"coeff (x^6 + 3*x^5 + 3*x^4 + 3*x^2 + 4*x + 2)/x^5\n",
|
||||
"(((-x^2 + x - 2)/y^5) dx, ((3*x^5 + 3*x^4 + 3*x^2 + 4*x + 2)/x^5)*y^2, 0 dx) 1\n",
|
||||
"c2\n",
|
||||
"c2\n",
|
||||
"c2c\n",
|
||||
"coeff (3*x^5 + 3*x^4 + 3*x^2 + 4*x + 2)/x^5\n",
|
||||
"((x/y^5) dx, ((3*x^4 + 3*x^2 + 4*x + 2)/x^5)*y^2, 0 dx) 1\n",
|
||||
"c2\n",
|
||||
"c2\n",
|
||||
"c2a\n",
|
||||
"coeff (3*x^4 + 3*x^2 + 4*x + 2)/x^5\n",
|
||||
"coeff 0\n",
|
||||
"3 0 2 1\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"ename": "ZeroDivisionError",
|
||||
"evalue": "inverse of Mod(0, 5) does not exist",
|
||||
"output_type": "error",
|
||||
"traceback": [
|
||||
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
|
||||
"\u001b[0;31mZeroDivisionError\u001b[0m Traceback (most recent call last)",
|
||||
"\u001b[0;32m<ipython-input-121-e1ff68010d76>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mC\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfrobenius_matrix\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<ipython-input-118-ac653a450b02>\u001b[0m in \u001b[0;36mfrobenius_matrix\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m 103\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mw\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mbasis\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mitems\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[1;32m 104\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'w'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mw\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 105\u001b[0;31m \u001b[0mv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mw\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfrobenius\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcoordinates\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 106\u001b[0m \u001b[0mM\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m:\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 107\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mM\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
|
||||
"\u001b[0;32m<ipython-input-118-ac653a450b02>\u001b[0m in \u001b[0;36mcoordinates\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m 414\u001b[0m \u001b[0ma\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcoeff_of_rational_fctn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfct_j\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 415\u001b[0m \u001b[0melt\u001b[0m \u001b[0;34m=\u001b[0m\u001b[0mself\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0msuperelliptic_cech\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdiffn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmult\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 416\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0melt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcoordinates\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 417\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 418\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mvector\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[0mg\u001b[0m\u001b[0;34m*\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[0;34m\u001b[0m\u001b[0m\n",
|
||||
"\u001b[0;32m<ipython-input-118-ac653a450b02>\u001b[0m in \u001b[0;36mcoordinates\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m 414\u001b[0m \u001b[0ma\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcoeff_of_rational_fctn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfct_j\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 415\u001b[0m \u001b[0melt\u001b[0m \u001b[0;34m=\u001b[0m\u001b[0mself\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0msuperelliptic_cech\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdiffn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmult\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 416\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0melt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcoordinates\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 417\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 418\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mvector\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[0mg\u001b[0m\u001b[0;34m*\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[0;34m\u001b[0m\u001b[0m\n",
|
||||
"\u001b[0;32m<ipython-input-118-ac653a450b02>\u001b[0m in \u001b[0;36mcoordinates\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m 414\u001b[0m \u001b[0ma\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcoeff_of_rational_fctn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfct_j\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 415\u001b[0m \u001b[0melt\u001b[0m \u001b[0;34m=\u001b[0m\u001b[0mself\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0msuperelliptic_cech\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdiffn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmult\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 416\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0melt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcoordinates\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 417\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 418\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mvector\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[0mg\u001b[0m\u001b[0;34m*\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[0;34m\u001b[0m\u001b[0m\n",
|
||||
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||||
"\u001b[0;32m/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/coerce.pyx\u001b[0m in \u001b[0;36msage.structure.coerce.CoercionModel.bin_op (build/cythonized/sage/structure/coerce.c:10143)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 1209\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_record_exception\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 1210\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[0;32m-> 1211\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mPyObject_CallObject\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mop\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mxy\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 1212\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 1213\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mop\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0mmul\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
|
||||
"\u001b[0;32m/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/element.pyx\u001b[0m in \u001b[0;36msage.structure.element.Element.__truediv__ (build/cythonized/sage/structure/element.c:12802)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 1714\u001b[0m \u001b[0mcdef\u001b[0m \u001b[0mint\u001b[0m \u001b[0mcl\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mclassify_elements\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mleft\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mright\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 1715\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mHAVE_SAME_PARENT\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcl\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-> 1716\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;34m<\u001b[0m\u001b[0mElement\u001b[0m\u001b[0;34m>\u001b[0m\u001b[0mleft\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_div_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mright\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 1717\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mBOTH_ARE_ELEMENT\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcl\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 1718\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mcoercion_model\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbin_op\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mleft\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mright\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtruediv\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
|
||||
"\u001b[0;32m/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/rings/finite_rings/integer_mod.pyx\u001b[0m in \u001b[0;36msage.rings.finite_rings.integer_mod.IntegerMod_int._div_ (build/cythonized/sage/rings/finite_rings/integer_mod.c:27689)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 2540\u001b[0m \u001b[0mright_inverse\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__modulus\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0minverses\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m<\u001b[0m\u001b[0mIntegerMod_int\u001b[0m\u001b[0;34m>\u001b[0m\u001b[0mright\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mivalue\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 2541\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mright_inverse\u001b[0m \u001b[0;32mis\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[0;32m-> 2542\u001b[0;31m \u001b[0;32mraise\u001b[0m \u001b[0mZeroDivisionError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34mf\"inverse of Mod({right}, {self.__modulus.sageInteger}) does not exist\"\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 2543\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 2544\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_new_c\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mivalue\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;34m<\u001b[0m\u001b[0mIntegerMod_int\u001b[0m\u001b[0;34m>\u001b[0m\u001b[0mright_inverse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mivalue\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__modulus\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mint32\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
|
||||
"\u001b[0;31mZeroDivisionError\u001b[0m: inverse of Mod(0, 5) does not exist"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"C.frobenius_matrix()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 4,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
@ -455,16 +666,17 @@
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"(((-x*y + 1)/y^5) dx, 4/x*y^4, ((-x^3*y + x*y^2 + x^2 - 2*y^2)/(x^2*y^5)) dx)\n",
|
||||
"((1/y^5) dx, 0, (1/y^5) dx)\n"
|
||||
"((1/y^5) dx, 0, (1/y^5) dx)\n",
|
||||
"(0 dx, 0, 0 dx)\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"(0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0)"
|
||||
"(0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0)"
|
||||
]
|
||||
},
|
||||
"execution_count": 118,
|
||||
"execution_count": 4,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
@ -850,6 +1062,55 @@
|
||||
"degrees1_inv"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 28,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"Superelliptic curve with the equation y^7 = x^3 + x + 2 over finite field with 5 elements."
|
||||
]
|
||||
},
|
||||
"execution_count": 28,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"C"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 29,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"basis = C.basis_de_rham()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 32,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"dict_items([(0, ((x/y) dx, 2/x*y, ((x^3*y^5 - x^3 + x - 1)/(x^2*y^6)) dx)), (1, (((-1)/y) dx, 2/x^2*y, ((-x^3*y^5 + x^3 - 2*x - 2)/(x^3*y^6)) dx)), (2, (((-2*x)/y^2) dx, 2/x*y^2, ((-2*x^3*y^3 + x^3 - 1)/(x^2*y^5)) dx)), (3, ((1/y^2) dx, 2/x^2*y^2, ((x^3*y^3 - 2*x^3 + 2*x - 2)/(x^3*y^5)) dx)), (4, ((1/y^3) dx, 0, (1/y^3) dx)), (5, (0 dx, 2/x*y^3, ((-2*x^3 - x - 1)/(x^2*y^4)) dx)), (6, ((1/y^4) dx, 0, (1/y^4) dx)), (7, ((2*x/y^4) dx, 2/x*y^4, ((2*x^3 - 2*x*y - y)/(x^2*y^4)) dx)), (8, ((1/y^5) dx, 0, (1/y^5) dx)), (9, ((x/y^5) dx, 0, (x/y^5) dx)), (10, ((1/y^6) dx, 0, (1/y^6) dx)), (11, ((x/y^6) dx, 0, (x/y^6) dx))])"
|
||||
]
|
||||
},
|
||||
"execution_count": 32,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"basis.items()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
|
||||
Loading…
Reference in New Issue
Block a user