diff --git a/superelliptic.ipynb b/superelliptic.ipynb index 6fb925f..f0ef55f 100644 --- a/superelliptic.ipynb +++ b/superelliptic.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 21, + "execution_count": 1, "metadata": {}, "outputs": [], "source": [ @@ -560,7 +560,7 @@ }, { "cell_type": "code", - "execution_count": 28, + "execution_count": 2, "metadata": {}, "outputs": [], "source": [ @@ -918,20 +918,21 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "[(1/y) dx, (x/y) dx, (x^2/y) dx, (x^3/y) dx]\n" + "[((1/y) dx, 0, (1/y) dx), ((x/y) dx, 2/x*y, (1/(x*y)) dx)]\n" ] } ], "source": [ - "C = superelliptic(x^9+x^8+x, 2, 11)\n", - "print(C.basis_holomorphic_differentials)" + "R. = PolynomialRing(GF(13))\n", + "C = superelliptic(x^3+x, 2, 13)\n", + "print(C.basis_de_rham)" ] }, { @@ -1885,6 +1886,38 @@ ], "source": [] }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Current Time = 18:56:38\n", + "[1, 2, 3, 4, 5, 6, 7, 8, 9]\n", + "Current Time = 18:57:11\n" + ] + } + ], + "source": [ + "now = datetime.now()\n", + "\n", + "current_time = now.strftime(\"%H:%M:%S\")\n", + "print(\"Current Time =\", current_time)\n", + "\n", + "p = 17\n", + "Rx. = PolynomialRing(GF(p))\n", + "C = superelliptic(x^19+x^8+x, 2, p)\n", + "print(C.final_type())\n", + "\n", + "now = datetime.now()\n", + "\n", + "current_time = now.strftime(\"%H:%M:%S\")\n", + "print(\"Current Time =\", current_time)" + ] + }, { "cell_type": "code", "execution_count": null, diff --git a/superelliptic_alpha.ipynb b/superelliptic_alpha.ipynb new file mode 100644 index 0000000..d4c8376 --- /dev/null +++ b/superelliptic_alpha.ipynb @@ -0,0 +1,1955 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [], + "source": [ + "def basis_holomorphic_differentials_degree(f, m, p):\n", + " kxi. = PolynomialRing(GF(p))\n", + " r = f.degree()\n", + " delta = GCD(r, m)\n", + " Rx. = PolynomialRing(kxi)\n", + " Rxy. = PolynomialRing(kxi, 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", + " kxi. = PolynomialRing(GF(p))\n", + " Rx. = PolynomialRing(kxi)\n", + " Rxy. = PolynomialRing(kxi, 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", + " kxi. = PolynomialRing(GF(p))\n", + " Rx. = PolynomialRing(kxi)\n", + " Rxy. = PolynomialRing(kxi, 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", + " kxi. = PolynomialRing(GF(p))\n", + " M = matrix(kxi, 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", + " kxi. = PolynomialRing(GF(p))\n", + " Rxy. = PolynomialRing(kxi, 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": {}, + "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": 8, + "metadata": {}, + "outputs": [], + "source": [ + "from datetime import datetime" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Current Time = 18:11:00\n", + "[1, 2, 3, 4, 5, 6]\n", + "Current Time = 18:11:21\n" + ] + } + ], + "source": [ + "now = datetime.now()\n", + "\n", + "current_time = now.strftime(\"%H:%M:%S\")\n", + "print(\"Current Time =\", current_time)\n", + "\n", + "p = 17\n", + "Rx. = PolynomialRing(GF(p))\n", + "C = superelliptic(x^13+x^8+x, 2, p)\n", + "print(C.final_type())\n", + "\n", + "now = datetime.now()\n", + "\n", + "current_time = now.strftime(\"%H:%M:%S\")\n", + "print(\"Current Time =\", current_time)" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "outputs": [], + "source": [ + "def p_cov(C):\n", + " m = C.exponent\n", + " p = C.characteristic\n", + " f = C.polynomial\n", + " return superelliptic(f(x^p - x), m, p)" + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "metadata": { + "scrolled": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[1, 1, 2, 3] [1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 10, 11]\n" + ] + } + ], + "source": [ + "p = 3\n", + "Rx. = PolynomialRing(GF(p))\n", + "C = superelliptic(2*x^9+x^8+x, 2, p)\n", + "C1 = p_cov(C)\n", + "print(C.final_type(), C1.final_type())" + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "( ['?', 1, 1, 2, 3, 3, 3, 4, 4] )\n", + "( ['?', '?', '?', '?', '?', 5, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, '?', 18, 19, 19, '?', 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 22, '?', '?', '?', '?', 22] )\n", + "[1, 1, 2, 3] [1, 2, 3, 4, 5, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 18, 18, 19]\n" + ] + } + ], + "source": [ + "p = 5\n", + "Rx. = PolynomialRing(GF(p))\n", + "C = superelliptic(x^9+x^8+x, 2, p)\n", + "C1 = p_cov(C)\n", + "print(C.final_type(1), C1.final_type(1))" + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "22" + ] + }, + "execution_count": 24, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "C1.genus()" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "x^9 + x^8 + x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + 2*x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + 3*x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + 4*x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + 5*x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + 6*x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + 7*x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + 8*x 3 0 0\n", + "(( [0, 0, 1, 2, 3, 3, 3, 3, 4] ))\n", + "[0, 1, 2, 3]\n", + "x^9 + x^8 + 9*x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + 2*x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + 3*x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + 4*x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + 5*x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + 6*x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + 7*x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + 8*x 3 0 0\n", + "(( [0, 0, 1, 2, 3, 3, 3, 3, 4] ))\n", + "[0, 1, 2, 3]\n", + "x^9 + x^8 + 9*x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + 2*x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + 3*x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + 4*x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + 5*x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + 6*x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + 7*x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + 8*x 3 0 0\n", + "(( [0, 0, 1, 2, 3, 3, 3, 3, 4] ))\n", + "[0, 1, 2, 3]\n", + "x^9 + x^8 + 9*x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + x 3 2 0\n", + "(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))\n", + "[1, 2, 2, 3]\n", + "x^9 + x^8 + 2*x 3 2 0\n" + ] + }, + { + "ename": "KeyboardInterrupt", + "evalue": "", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mKeyboardInterrupt\u001b[0m Traceback (most recent call last)", + "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[1;32m 9\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpolynomial\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mp_rank\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mM\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m4\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrank\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m4\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mM\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0midentity_matrix\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m4\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrank\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 10\u001b[0m \u001b[0mV\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mverschiebung_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[0;32m---> 11\u001b[0;31m \u001b[0mF\u001b[0m \u001b[0;34m=\u001b[0m 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None\"\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mformat\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmethod_name\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtype\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[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/polynomial/multi_polynomial.pyx\u001b[0m in \u001b[0;36msage.rings.polynomial.multi_polynomial.MPolynomial._polynomial_ (build/cythonized/sage/rings/polynomial/multi_polynomial.c:6311)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 205\u001b[0m \u001b[0mvar\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mR\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mvariable_name\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 206\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mvar\u001b[0m \u001b[0;32min\u001b[0m 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\u001b[0;36msage.structure.coerce_maps.DefaultConvertMap_unique._call_ (build/cythonized/sage/structure/coerce_maps.c:4448)\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 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"\u001b[0;31mKeyboardInterrupt\u001b[0m: " + ] + } + ], + "source": [ + "p = 11\n", + "for a in range(0, p):\n", + " for b in range(0, p):\n", + " for c in range(0, p):\n", + " Rx. = PolynomialRing(GF(p))\n", + " C = superelliptic(x^9+a*x^8+c*x, 2, p)\n", + " if C.is_smooth() and C.p_rank() == 3:\n", + " M = C.cartier_matrix()\n", + " print(C.polynomial, C.p_rank(), (M^4).rank(), 4 - (M-identity_matrix(4)).rank())\n", + " V = C.verschiebung_matrix()\n", + " F = C.frobenius_matrix()\n", + " print(flag(F, V, p))" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[((1/y) dx, 0, (1/y) dx), ((x/y) dx, 2/x*y, (1/(x*y)) dx)]\n" + ] + } + ], + "source": [ + "R. = PolynomialRing(GF(13))\n", + "C = superelliptic(x^3+x, 2, 13)\n", + "print(C.basis_de_rham)" + ] + }, + { + "cell_type": "code", + "execution_count": 81, + "metadata": { + "scrolled": true + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "1\n", + "[1]\n" + ] + } + ], + "source": [ + "p = 5\n", + "Rx. = PolynomialRing(GF(p))\n", + "C = superelliptic(x^3 - x, 2, p)\n", + "if (C.is_smooth()):\n", + " print(C.p_rank())\n", + " print(C.final_type())" + ] + }, + { + "cell_type": "code", + "execution_count": 58, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[0 0 0 0 0 0 0 0]\n", + "[0 0 0 0 0 0 0 0]\n", + "[0 0 0 0 0 0 0 0]\n", + "[0 0 0 0 0 0 0 0]\n", + "[0 0 0 4 2 3 3 2]\n", + "[4 0 4 4 2 0 0 4]\n", + "[0 0 4 3 0 0 1 2]\n", + "[3 0 0 0 0 1 0 0]" + ] + }, + "execution_count": 58, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "C.frobenius_matrix()" + ] + }, + { + "cell_type": "code", + "execution_count": 62, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "test [0, -1, -1, -1, 4, -1, -1, -1, 4]\n", + "poprawiony ft [0, 1, 2, 3, 4, 4, 4, 4, 4]\n" + ] + }, + { + "data": { + "text/plain": [ + "[1, 2, 3, 4]" + ] + }, + "execution_count": 62, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "flag(F, V, p)" + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "metadata": { + "scrolled": true + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "4*x^9 + 4*x^3 + 2*x^2 + x\n", + "4*x^9 + 2*x^3 + 4*x^2 + 4*x + 4\n" + ] + } + ], + "source": [ + "f = x^9+3*x^3+x^2+x+1\n", + "r = f.degree()\n", + "i = 2\n", + "j = 1\n", + "m = 2\n", + "print(Rx(m-j)*Rx(x)*Rx(f.derivative()))\n", + "print(Rx(m)*Rx(i)*f)" + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "9\n", + "4\n", + "4*x^8 + 4*x^2 + 2*x + 1\n" + ] + } + ], + "source": [ + "print(r*(m-j))\n", + "print(m*i)\n", + "print(f.derivative())" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "2*x^3 + 3*x^2 + 2*x + 1" + ] + }, + "execution_count": 31, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "x*f.derivative() - 2*2*f" + ] + }, + { + "cell_type": "code", + "execution_count": 73, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "Vector space of degree 2 and dimension 1 over Rational Field\n", + "Basis matrix:\n", + "[1 3]" + ] + }, + "execution_count": 73, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "M = matrix(QQ, [[1,2], [3,6]])\n", + "U = M.kernel()\n", + "V = VectorSpace(QQ,2)\n", + "M.transpose().image()" + ] + }, + { + "cell_type": "code", + "execution_count": 77, + "metadata": {}, + "outputs": [], + "source": [ + "l = U.basis()\n", + "l = l +[(1, 1/3)]" + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "metadata": {}, + "outputs": [], + "source": [ + "###fragment kodu do obliczania residuuow w niesk - zaniechany\n", + "#def more_v(f, prec):\n", + "# C = f.curve\n", + "# f = f.vw\n", + "# g = C.polynomial8\n", + "# p = C.characteristic\n", + "# m = C.exponent\n", + "# r = C.polynomial.degree()\n", + "# delta, a, b = xgcd(m, r)\n", + "# a = -a\n", + "# M = m/delta\n", + "# R = r/delta\n", + "# \n", + "# Fpbar = GF(p).algebraic_closure()\n", + "# Ruv. = PolynomialRing(Fpbar, 2)\n", + "# if prec == 0:\n", + "# return 0\n", + "# zeta = Fpbar.zeta(m)\n", + "# a = f(v = zeta, w = 0)\n", + "# f1 = f - a\n", + "# if w.divides(f1):\n", + "# return more_v(f1/w, prec-1)\n", + " " + ] + }, + { + "cell_type": "code", + "execution_count": 211, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "4\n", + "2\n" + ] + } + ], + "source": [ + "p = 7\n", + "Rx. = PolynomialRing(GF(p))\n", + "C = superelliptic(x^3 + x + 3, 5, p)\n", + "baza = C.basis_de_rham\n", + "print(C.genus())\n", + "#E = EllipticCurve(GF(p), [1, 2])\n", + "print(E.trace_of_frobenius())\n", + "#C.basis_holomorphic_differentials( )" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": 212, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "{0: (0, 2),\n", + " 1: (0, 3),\n", + " 2: (0, 4),\n", + " 3: (1, 4),\n", + " 4: (-1, 1),\n", + " 5: (0, 1),\n", + " 6: (1, 2),\n", + " 7: (1, 3)}" + ] + }, + "execution_count": 212, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "C.degree_de_rham0" + ] + }, + { + "cell_type": "code", + "execution_count": 213, + "metadata": { + "scrolled": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "{0: ((1/y^2) dx, 0, (1/y^2) dx),\n", + " 1: ((1/y^3) dx, 0, (1/y^3) dx),\n", + " 2: ((1/y^4) dx, 0, (1/y^4) dx),\n", + " 3: ((x/y^4) dx, 0, (x/y^4) dx),\n", + " 4: (0 dx, 5/x*y^4, ((x + 1)/(x^2*y)) dx),\n", + " 5: ((2/y) dx, 5/x^2*y^4, ((-x + 2)/(x^3*y)) dx),\n", + " 6: (((-3*x)/y^2) dx, 5/x*y^3, ((2*x + 1)/(x^2*y^2)) dx),\n", + " 7: ((x/y^3) dx, 5/x*y^2, ((3*x + 1)/(x^2*y^3)) dx)}" + ] + }, + "execution_count": 213, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "C.basis_de_rham" + ] + }, + { + "cell_type": "code", + "execution_count": 214, + "metadata": {}, + "outputs": [], + "source": [ + "A = C.frobenius_matrix()\n", + "B = C.verschiebung_matrix()" + ] + }, + { + "cell_type": "code", + "execution_count": 227, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(1, 0, 0, 4, 3, 0, 2, 0)" + ] + }, + "execution_count": 227, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "A.solve_right((0, 0, 0, 0, 4, 2, 5, 3))" + ] + }, + { + "cell_type": "code", + "execution_count": 228, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(0, 0, 0, 0, 4, 2, 5, 3)" + ] + }, + "execution_count": 228, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "A*vector((1, 0, 0, 4, 3, 0, 2, 0))" + ] + }, + { + "cell_type": "code", + "execution_count": 225, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(1, 0, 0, 4, 3, -6, -5, -5)" + ] + }, + "execution_count": 225, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "vector((1, 0, 0, 4, 5, 0, 0, 0)) - vector((0, 0, 0, 0, 2, 6, 5, 5))" + ] + }, + { + "cell_type": "code", + "execution_count": 83, + "metadata": {}, + "outputs": [], + "source": [ + "omega = diffn(superelliptic_function(C, y^2))" + ] + }, + { + "cell_type": "code", + "execution_count": 84, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "3*x^2 + 1" + ] + }, + "execution_count": 84, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "omega.jth_component(0)" + ] + }, + { + "cell_type": "code", + "execution_count": 85, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "y" + ] + }, + "execution_count": 85, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "R. = PolynomialRing(GF(p), 2)\n", + "g1 = x^3*y^7 + x^2*y^9\n", + "g2 = x^2*y + y^6\n", + "R1. = PolynomialRing(GF(p))\n", + "R2 = FractionField(R1)\n", + "R3. = PolynomialRing(R2)\n", + "\n", + "xgcd(R3(g1), R3(g2))[1]*R3(g1) + xgcd(R3(g1), R3(g2))[2]*R3(g2)" + ] + }, + { + "cell_type": "code", + "execution_count": 86, + "metadata": {}, + "outputs": [], + "source": [ + "H = HyperellipticCurve(x^5 - x + 1)" + ] + }, + { + "cell_type": "code", + "execution_count": 40, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "Hyperelliptic Curve over Finite Field of size 5 defined by y^2 = x^5 + 4*x + 1" + ] + }, + "execution_count": 40, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "H" + ] + }, + { + "cell_type": "code", + "execution_count": 84, + "metadata": {}, + "outputs": [], + "source": [ + "f = x^3 + x + 2" + ] + }, + { + "cell_type": "code", + "execution_count": 86, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "-2*x^2 + 1" + ] + }, + "execution_count": 86, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "f.derivative(x)" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "p = 5\n", + "R1. = PolynomialRing(GF(p))\n", + "R2 = FractionField(R1)\n", + "R3. = PolynomialRing(R2)\n", + "g = y^2/x + y/(x+1) \n", + "g = 1/y+x/y^2" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "x*z^2 + z" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "R3. = PolynomialRing(R2)\n", + "g(y = 1/z)" + ] + }, + { + "cell_type": "code", + "execution_count": 57, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "x^3 + x + 4" + ] + }, + "execution_count": 57, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "f" + ] + }, + { + "cell_type": "code", + "execution_count": 62, + "metadata": {}, + "outputs": [ + { + "ename": "AttributeError", + "evalue": "'sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint' object has no attribute 'coefficient'", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mAttributeError\u001b[0m Traceback (most recent call last)", + "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcoefficient\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/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/element.pyx\u001b[0m in \u001b[0;36msage.structure.element.Element.__getattr__ (build/cythonized/sage/structure/element.c:4614)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 485\u001b[0m \u001b[0mAttributeError\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0;34m'LeftZeroSemigroup_with_category.element_class'\u001b[0m \u001b[0mobject\u001b[0m \u001b[0mhas\u001b[0m \u001b[0mno\u001b[0m \u001b[0mattribute\u001b[0m \u001b[0;34m'blah_blah'\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 486\u001b[0m \"\"\"\n\u001b[0;32m--> 487\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 488\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 489\u001b[0m \u001b[0mcdef\u001b[0m \u001b[0mgetattr_from_category\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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[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.getattr_from_category (build/cythonized/sage/structure/element.c:4723)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 498\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 499\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--> 500\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 501\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 502\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0m__dir__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/cpython/getattr.pyx\u001b[0m in \u001b[0;36msage.cpython.getattr.getattr_from_other_class (build/cythonized/sage/cpython/getattr.c:2614)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 392\u001b[0m \u001b[0mdummy_error_message\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcls\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 393\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--> 394\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 395\u001b[0m \u001b[0mattribute\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m<\u001b[0m\u001b[0mobject\u001b[0m\u001b[0;34m>\u001b[0m\u001b[0mattr\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 396\u001b[0m \u001b[0;31m# Check for a descriptor (__get__ in Python)\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.polynomial_zmod_flint.Polynomial_zmod_flint' object has no attribute 'coefficient'" + ] + } + ], + "source": [ + "f.coefficient()" + ] + }, + { + "cell_type": "code", + "execution_count": 35, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "x^3 + x + 1" + ] + }, + "execution_count": 35, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "x^3+x+1" + ] + }, + { + "cell_type": "code", + "execution_count": 36, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "Symbolic Ring" + ] + }, + "execution_count": 36, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "parent(x)" + ] + }, + { + "cell_type": "code", + "execution_count": 37, + "metadata": {}, + "outputs": [], + "source": [ + "R. = PolynomialRing(GF(5))" + ] + }, + { + "cell_type": "code", + "execution_count": 39, + "metadata": {}, + "outputs": [], + "source": [ + "R = (x^3+x).parent()" + ] + }, + { + "cell_type": "code", + "execution_count": 44, + "metadata": {}, + "outputs": [], + "source": [ + "R. = PolynomialRing(GF(5))\n", + "RR = FractionField(R)\n", + "A = RR(1/(x*y))" + ] + }, + { + "cell_type": "code", + "execution_count": 45, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(-1)/(x^2*y)" + ] + }, + "execution_count": 45, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "A.derivative(x)" + ] + }, + { + "cell_type": "code", + "execution_count": 42, + "metadata": {}, + "outputs": [], + "source": [ + "dict1 = {}\n", + "dict1[3] = 5\n", + "dict1[6] = 121" + ] + }, + { + "cell_type": "code", + "execution_count": 46, + "metadata": {}, + "outputs": [], + "source": [ + "degrees1_inv = {b:a for a, b in dict1.items()}" + ] + }, + { + "cell_type": "code", + "execution_count": 47, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "{5: 3, 121: 6}" + ] + }, + "execution_count": 47, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "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": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(3, 1, -1)" + ] + }, + "execution_count": 9, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "m = 9\n", + "r = 6\n", + "delta, a, b = xgcd(m, r)\n", + "a = -a\n", + "xgcd(9, 6)" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(3, 3)" + ] + }, + "execution_count": 10, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "b*r -a*m, delta" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [], + "source": [ + "Fpbar = GF(5).algebraic_closure()\n", + "z = Fpbar.zeta(7)" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": {}, + "outputs": [ + { + "ename": "TypeError", + "evalue": "unable to coerce ", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mTypeError\u001b[0m Traceback (most recent call last)", + "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mGF\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m5\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m6\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mz\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/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/parent.pyx\u001b[0m in \u001b[0;36msage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9218)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 898\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 899\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--> 900\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 901\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 902\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/opt/sagemath-9.1/local/lib/python3.7/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:4556)\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/opt/sagemath-9.1/local/lib/python3.7/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:4448)\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/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/rings/finite_rings/finite_field_givaro.py\u001b[0m in \u001b[0;36m_element_constructor_\u001b[0;34m(self, e)\u001b[0m\n\u001b[1;32m 368\u001b[0m \u001b[0;36m2\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0ma4\u001b[0m\u001b[0;34m^\u001b[0m\u001b[0;36m3\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0;36m2\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0ma4\u001b[0m\u001b[0;34m^\u001b[0m\u001b[0;36m2\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 369\u001b[0m \"\"\"\n\u001b[0;32m--> 370\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_cache\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0melement_from_data\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0me\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 371\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 372\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mgen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mn\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/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/rings/finite_rings/element_givaro.pyx\u001b[0m in \u001b[0;36msage.rings.finite_rings.element_givaro.Cache_givaro.element_from_data (build/cythonized/sage/rings/finite_rings/element_givaro.cpp:7458)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 312\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mmake_FiniteField_givaroElement\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mres\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 313\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 314\u001b[0;31m \u001b[0mcpdef\u001b[0m \u001b[0mFiniteField_givaroElement\u001b[0m \u001b[0melement_from_data\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0me\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 315\u001b[0m \"\"\"\n\u001b[1;32m 316\u001b[0m \u001b[0mCoerces\u001b[0m \u001b[0mseveral\u001b[0m \u001b[0mdata\u001b[0m \u001b[0mtypes\u001b[0m \u001b[0mto\u001b[0m\u001b[0;31m \u001b[0m\u001b[0;31m`\u001b[0m\u001b[0;31m`\u001b[0m\u001b[0mself\u001b[0m\u001b[0;31m`\u001b[0m\u001b[0;31m`\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/element_givaro.pyx\u001b[0m in \u001b[0;36msage.rings.finite_rings.element_givaro.Cache_givaro.element_from_data (build/cythonized/sage/rings/finite_rings/element_givaro.cpp:7080)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 451\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 452\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--> 453\u001b[0;31m \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"unable to coerce %r\"\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0me\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 454\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 455\u001b[0m \u001b[0mcdef\u001b[0m \u001b[0mGEN\u001b[0m \u001b[0mt\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;31mTypeError\u001b[0m: unable to coerce " + ] + } + ], + "source": [ + "GF(5^6)(z)" + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "6" + ] + }, + "execution_count": 21, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Integers(7)(5).multiplicative_order()" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [], + "source": [ + "Rx. = PolynomialRing(QQ)\n", + "f = sum((i+1)*x^i for i in range(0, 10))" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "10*x^3 + 9*x^2 + 8*x + 7" + ] + }, + "execution_count": 8, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "cut(f, 5)" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "10*x^9 + 9*x^8 + 8*x^7 + 7*x^6 + 6*x^5 + 5*x^4 + 4*x^3 + 3*x^2 + 2*x + 1" + ] + }, + "execution_count": 9, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "f" + ] + }, + { + "cell_type": "code", + "execution_count": 45, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Elliptic Curve defined by y^2 + y = x^3 + 1 over Finite Field of size 2\n", + "x^2 + 2\n" + ] + } + ], + "source": [ + "E = EllipticCurve(GF(2), [0,0,1,0,1])\n", + "print(E)\n", + "print(E.frobenius_polynomial())" + ] + }, + { + "cell_type": "code", + "execution_count": 41, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "Elliptic Curve defined by y^2 + y = x^3 + x over Finite Field of size 2" + ] + }, + "execution_count": 41, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "E" + ] + }, + { + "cell_type": "code", + "execution_count": 42, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "x^2 + 2*x + 2" + ] + }, + "execution_count": 42, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Current Time = 18:56:38\n", + "[1, 2, 3, 4, 5, 6, 7, 8, 9]\n", + "Current Time = 18:57:11\n" + ] + } + ], + "source": [ + "now = datetime.now()\n", + "\n", + "current_time = now.strftime(\"%H:%M:%S\")\n", + "print(\"Current Time =\", current_time)\n", + "\n", + "p = 17\n", + "Rx. = PolynomialRing(GF(p))\n", + "C = superelliptic(x^19+x^8+x, 2, p)\n", + "print(C.final_type())\n", + "\n", + "now = datetime.now()\n", + "\n", + "current_time = now.strftime(\"%H:%M:%S\")\n", + "print(\"Current Time =\", current_time)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "SageMath 9.1", + "language": "sage", + "name": "sagemath" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.7.3" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +}