{ "cells": [ { "cell_type": "code", "execution_count": 118, "metadata": {}, "outputs": [], "source": [ "class superelliptic:\n", " def __init__(self, f, m, p):\n", " Rx. = PolynomialRing(GF(p))\n", " self.polynomial = Rx(f)\n", " self.exponent = m\n", " self.characteristic = p\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 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 basis_holomorphic_differentials(self, j = 'all'):\n", " f = self.polynomial\n", " m = self.exponent\n", " p = self.characteristic\n", " r = f.degree()\n", " delta = GCD(r, m)\n", " Rxy. = PolynomialRing(GF(p), 2)\n", " Fxy = FractionField(Rxy)\n", " \n", " basis = {}\n", " k = 0\n", " if j == 'all':\n", " for j in range(1, m):\n", " for i in range(1, r):\n", " if (r*j - m*i >= delta):\n", " basis[k] = superelliptic_form(self, Fxy(x^(i-1)/y^j))\n", " k = k+1\n", " return basis\n", " else:\n", " for i in range(1, r):\n", " if (r*j - m*i >= delta):\n", " basis[k] = superelliptic_form(self, Fxy(x^(i-1)/y^j))\n", " k = k+1\n", " return basis\n", " \n", " def degree_and_basis_de_rham(self, j = 'all'):\n", " f = self.polynomial\n", " m = self.exponent\n", " p = self.characteristic\n", " r = f.degree()\n", " delta = GCD(r, m)\n", " Rx. = PolynomialRing(GF(p))\n", " Rxy. = PolynomialRing(GF(p), 2)\n", " Fxy = FractionField(Rxy)\n", " basis = {}\n", " degrees0 = {}\n", " degrees1 = {}\n", " t = 0\n", " if j == 'all':\n", " for j in range(1, m):\n", " holo = C.basis_holomorphic_differentials(j)\n", " for k in range(0, len(holo)):\n", " basis[t] = superelliptic_cech(self, holo[k], superelliptic_function(self, Rx(0))) \n", " g = Rx(holo[k].jth_component(j))\n", " degrees0[t] = (g.degree(), j)\n", " t += 1\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[t] = superelliptic_cech(self, superelliptic_form(self, Fxy(psi/y^j)), superelliptic_function(self, Fxy(m*y^(m-j)/x^i)))\n", " degrees0[t] = (psi.degree(), j)\n", " degrees1[t] = (-i, j)\n", " t += 1\n", " return basis, degrees0, degrees1\n", " \n", " def degree_de_rham(self, i, j='all'):\n", " basis, degrees0, degrees1 = self.degree_and_basis_de_rham(j)\n", " if i==0:\n", " return degrees0\n", " \n", " if i==1:\n", " return degrees1\n", " \n", " def basis_de_rham(self, j = 'all'): \n", " basis, degrees0, degrees1 = self.degree_and_basis_de_rham(j)\n", " return basis\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, w in basis.items():\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", " for i, w in basis.items():\n", " print('w', w)\n", " v = w.frobenius().coordinates()\n", " M[i, :] = v\n", " return M\n", " \n", "def reduction(C, g):\n", " p = C.characteristic\n", " Rxy. = PolynomialRing(GF(p), 2)\n", " Fxy = FractionField(Rxy)\n", " f = C.polynomial\n", " r = f.degree()\n", " m = C.exponent\n", " g = Fxy(g)\n", " g1 = g.numerator()\n", " g2 = g.denominator()\n", " \n", " Rx. = PolynomialRing(GF(p))\n", " Fx = FractionField(Rx)\n", " FxRy. = PolynomialRing(Fx) \n", " (A, B, C) = xgcd(FxRy(g2), FxRy(y^m - f))\n", " g = FxRy(g1*B/A)\n", " \n", " while(g.degree(Rxy(y)) >= m):\n", " d = g.degree(Rxy(y))\n", " G = coff(g, d)\n", " i = floor(d/m)\n", " g = g - G*y^d + f^i * y^(d%m) *G\n", " \n", " return(FxRy(g))\n", "\n", "def reduction_form(C, g):\n", " p = C.characteristic\n", " Rxy. = PolynomialRing(GF(p), 2)\n", " Fxy = FractionField(Rxy)\n", " f = C.polynomial\n", " r = f.degree()\n", " m = C.exponent\n", " g = reduction(C, g)\n", "\n", " g1 = Rxy(0)\n", " Rx. = PolynomialRing(GF(p))\n", " Fx = FractionField(Rx)\n", " FxRy. = PolynomialRing(Fx)\n", " \n", " g = FxRy(g)\n", " for j in range(0, m):\n", " if j==0:\n", " G = coff(g, 0)\n", " g1 += FxRy(G)\n", " else:\n", " G = coff(g, j)\n", " g1 += Fxy(y^(j-m)*f*G)\n", " return(g1)\n", " \n", "class superelliptic_function:\n", " def __init__(self, C, g):\n", " p = C.characteristic\n", " Rxy. = PolynomialRing(GF(p), 2)\n", " Fxy = FractionField(Rxy)\n", " f = C.polynomial\n", " r = f.degree()\n", " m = C.exponent\n", " \n", " self.curve = C\n", " g = reduction(C, g)\n", " self.function = g\n", " \n", " def __repr__(self):\n", " return str(self.function)\n", " \n", " def jth_component(self, j):\n", " g = self.function\n", " C = self.curve\n", " p = C.characteristic\n", " Rx. = PolynomialRing(GF(p))\n", " Fx. = FractionField(Rx)\n", " FxRy. = PolynomialRing(Fx)\n", " g = FxRy(g)\n", " return coff(g, j)\n", " \n", " def __add__(self, other):\n", " C = self.curve\n", " g1 = self.function\n", " g2 = other.function\n", " g = reduction(C, g1 + g2)\n", " return superelliptic_function(C, g)\n", " \n", " def __sub__(self, other):\n", " C = self.curve\n", " g1 = self.function\n", " g2 = other.function\n", " g = reduction(C, g1 - g2)\n", " return superelliptic_function(C, g)\n", " \n", " def __mul__(self, other):\n", " C = self.curve\n", " g1 = self.function\n", " g2 = other.function\n", " g = reduction(C, g1 * g2)\n", " return superelliptic_function(C, g)\n", " \n", " def __truediv__(self, other):\n", " C = self.curve\n", " g1 = self.function\n", " g2 = other.function\n", " g = reduction(C, g1 / g2)\n", " return superelliptic_function(C, g)\n", " \n", "def diffn(self):\n", " C = self.curve\n", " f = C.polynomial\n", " m = C.exponent\n", " p = C.characteristic\n", " g = self.function\n", " Rxy. = PolynomialRing(GF(p), 2)\n", " Fxy = FractionField(Rxy)\n", " g = Fxy(g)\n", " A = g.derivative(x)\n", " B = g.derivative(y)*f.derivative(x)/(m*y^(m-1))\n", " return superelliptic_form(C, A+B)\n", " \n", "class superelliptic_form:\n", " def __init__(self, C, g):\n", " p = C.characteristic\n", " Rxy. = PolynomialRing(GF(p), 2)\n", " Fxy = FractionField(Rxy)\n", " g = Fxy(reduction_form(C, g))\n", " self.form = g\n", " self.curve = C \n", " \n", " def __add__(self, other):\n", " C = self.curve\n", " g1 = self.form\n", " g2 = other.form\n", " g = reduction(C, g1 + g2)\n", " return superelliptic_form(C, g)\n", " \n", " def __sub__(self, other):\n", " C = self.curve\n", " g1 = self.form\n", " g2 = other.form\n", " g = reduction(C, g1 - g2)\n", " return superelliptic_form(C, g)\n", " \n", " def __repr__(self):\n", " g = self.form\n", " if len(str(g)) == 1:\n", " return str(g) + ' dx'\n", " return '('+str(g) + ') dx'\n", " \n", " def 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 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)\n", " if(-d*M + j*R -(M+1)<0):\n", " print('not', d, j, r, m, -d*M + j*R -(M+1))\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 mult(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", " print(self, self.is_cocycle())\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", " degrees0 = C.degree_de_rham(0)\n", " degrees0_inv = {b:a for a, b in degrees0.items()} \n", " degrees1 = C.degree_de_rham(1)\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", " print('c1')\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)\n", " index = degrees0_inv[(d, j)]\n", " print('baza', basis[index])\n", " a = coeff_of_rational_fctn(omega_j)\n", " a1 = coeff_of_rational_fctn(basis[index].omega0.jth_component(j))\n", " elt = self - basis[index].mult(a/a1)\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", " print('c2')\n", " fct_j = Fx(fct.jth_component(j))\n", " if (fct_j != Rx(0)):\n", " d = degree_of_rational_fctn(fct_j)\n", " \n", " if (d, j) in degrees1.values():\n", " print('c2a')\n", " index = degrees1_inv[(d, j)]\n", " a = coeff_of_rational_fctn(fct_j)\n", " a1 = coeff_of_rational_fctn(basis[index].f.jth_component(j))\n", " print(a, a1, index, basis[index].is_cocycle())\n", " elt = self - basis[index].mult(a/a1)\n", " return elt.coordinates() + a/a1*vector([GF(p)(i == index) for i in range(0, 2*g)])\n", " \n", " if d<0:\n", " print('c2b')\n", " a = coeff_of_rational_fctn(fct_j)\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", " print('c2c')\n", " G = superelliptic_function(C, y^j*x^d)\n", " a = coeff_of_rational_fctn(fct_j)\n", " elt =self - superelliptic_cech(C, diffn(G), G).mult(a)\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):\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):\n", " print('coeff', f)\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": 119, "metadata": {}, "outputs": [], "source": [ "p = 5\n", "C = superelliptic(x^3 + x + 2, 7, p)\n", "baza = C.basis_de_rham()\n", "#C.basis_holomorphic_differentials()" ] }, { "cell_type": "code", "execution_count": 120, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "((x/y) dx, 2/x*y^6, ((x - 1)/(x^2*y)) dx) 1\n", "(((-1)/y) dx, 2/x^2*y^6, ((-2*x - 2)/(x^3*y)) dx) 1\n", "(((-2*x)/y^2) dx, 2/x*y^5, ((2*x - 1)/(x^2*y^2)) dx) 1\n", "((1/y^2) dx, 2/x^2*y^5, ((-x - 2)/(x^3*y^2)) dx) 1\n", "((1/y^3) dx, 0, (1/y^3) dx) 1\n", "(0 dx, 2/x*y^4, ((-2*x - 1)/(x^2*y^3)) dx) 1\n", "((1/y^4) dx, 0, (1/y^4) dx) 1\n", "((2*x/y^4) dx, 2/x*y^3, ((-x - 1)/(x^2*y^4)) dx) 1\n", "((1/y^5) dx, 0, (1/y^5) dx) 1\n", "((x/y^5) dx, 0, (x/y^5) dx) 1\n", "((1/y^6) dx, 0, (1/y^6) dx) 1\n", "((x/y^6) dx, 0, (x/y^6) dx) 1\n" ] } ], "source": [ "for w in baza.values():\n", " print(w, w.is_cocycle())" ] }, { "cell_type": "code", "execution_count": 75, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "({0: ((x/y) dx, 2/x*y, ((x^3*y^5 - x^3 + x - 1)/(x^2*y^6)) dx),\n", " 1: (((-1)/y) dx, 2/x^2*y, ((-x^3*y^5 + x^3 - 2*x - 2)/(x^3*y^6)) dx),\n", " 2: (((-2*x)/y^2) dx, 2/x*y^2, ((-2*x^3*y^3 + x^3 - 1)/(x^2*y^5)) dx),\n", " 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\u001b[0m in \u001b[0;36m\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\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 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\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": [ { "name": "stdout", "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", "(0 dx, 0, 0 dx)\n" ] }, { "data": { "text/plain": [ "(0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0)" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "w.coordinates()" ] }, { "cell_type": "code", "execution_count": 65, "metadata": {}, "outputs": [], "source": [ "a, b, c = C.degree_and_basis_de_rham()" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "ename": "NameError", "evalue": "name 'y' is not defined", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mNameError\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 8\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m 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\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m/\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0mj\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 11\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0momega\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_regular_on_U0\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0momega\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_regular_on_Uinfty\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[1;32m 12\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0momega\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;31mNameError\u001b[0m: name 'y' is not defined" ] } ], "source": [ "licz = 0\n", "m = 2\n", "p = 5\n", "R1. = PolynomialRing(GF(p))\n", "f = R1(x^3 + x + 4)\n", "r = f.degree()\n", "C = superelliptic(f, m, p)\n", "for i in range(0, r):\n", " for j in range(1, m):\n", " omega = superelliptic_form(C, x^i/y^j)\n", " if (omega.is_regular_on_U0() and omega.is_regular_on_Uinfty()):\n", " print(omega)\n", " licz += 1\n", "print(licz, C.genus())\n", "print(C.basis_holomorphic_differentials())" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "p = 5\n", "R. = PolynomialRing(GF(p), 2)\n", "g = x^6*y^2 + y^2" ] }, { "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": 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 }