420 lines
10 KiB
Plaintext
420 lines
10 KiB
Plaintext
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 219,
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"metadata": {},
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"outputs": [],
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"source": [
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"class superelliptic:\n",
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" def __init__(self, f, m, p):\n",
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" R.<x> = PolynomialRing(GF(p))\n",
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" self.polynomial = R(f)\n",
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" self.exponent = m\n",
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" self.characteristic = p\n",
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" \n",
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" \n",
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" def __repr__(self):\n",
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" f = self.polynomial\n",
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" m = self.exponent\n",
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" p = self.characteristic\n",
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" return 'Superelliptic curve with the equation y^' + str(m) + ' = ' + str(f)+' over finite field with ' + str(p) + ' elements.'\n",
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" \n",
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" def genus(self):\n",
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" r = self.polynomial.degree()\n",
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" m = self.exponent\n",
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" delta = GCD(r, m)\n",
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" return 1/2*((r-1)*(m-1) - delta + 1)\n",
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" \n",
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" def basis_holomorphic_differentials(self, j = 'all'):\n",
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" f = self.polynomial\n",
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" m = self.exponent\n",
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" p = self.characteristic\n",
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" r = f.degree()\n",
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" delta = GCD(r, m)\n",
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" \n",
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" basis = {}\n",
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" if j == 'all':\n",
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" k = 0\n",
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" for i in range(1, r):\n",
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" for j in range(1, m):\n",
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" if (r*j - m*i >= delta):\n",
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" basis[k] = superelliptic_form(C, x^(i-1)/y^j)\n",
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" k = k+1\n",
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" return basis\n",
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" else:\n",
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" k = 0\n",
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" for i in range(1, r):\n",
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" if (r*j - m*i >= delta):\n",
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" basis[k] = superelliptic_form(C, x^(i-1)/y^j)\n",
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" k = k+1\n",
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" return basis\n",
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" \n",
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"def reduction(C, g):\n",
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" p = C.characteristic\n",
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" R.<x, y> = PolynomialRing(GF(p), 2)\n",
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" RR = FractionField(R)\n",
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" f = C.polynomial\n",
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" r = f.degree()\n",
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" m = C.exponent\n",
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" g = RR(g)\n",
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" g1 = g.numerator()\n",
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" g2 = g.denominator()\n",
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" \n",
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" R1.<x> = PolynomialRing(GF(p))\n",
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" R2 = FractionField(R1)\n",
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" R3.<y> = PolynomialRing(R2) \n",
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" (A, B, C) = xgcd(R3(g2), R3(y^m - f))\n",
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" g = R3(g1*B/A)\n",
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" \n",
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" while(g.degree(R(y)) >= m):\n",
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" d = g.degree(R(y))\n",
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" G = g.coefficient(R(y^d))\n",
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" i = floor(d/m)\n",
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" g = g - G*y^d + f^i * y^(d%m) *G\n",
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" \n",
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" return(R3(g))\n",
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"\n",
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"def reduction_form(C, g):\n",
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" p = C.characteristic\n",
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" R.<x, y> = PolynomialRing(GF(p), 2)\n",
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" RR = FractionField(R)\n",
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" f = C.polynomial\n",
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" r = f.degree()\n",
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" m = C.exponent\n",
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" g = reduction(C, g)\n",
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"\n",
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" g1 = RR(0)\n",
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" R1.<x> = PolynomialRing(GF(p))\n",
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" R2 = FractionField(R1)\n",
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" R3.<y> = PolynomialRing(R2)\n",
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" \n",
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" g = R3(g)\n",
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" for j in range(0, m):\n",
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" G = g.coefficients(sparse = false)[j]\n",
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" g1 += RR(y^(j-m)*f*G)\n",
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" \n",
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" return(g1)\n",
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" \n",
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"class superelliptic_function:\n",
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" def __init__(self, C, g):\n",
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" R.<x, y> = PolynomialRing(GF(p), 2)\n",
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" RR = FractionField(R)\n",
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" f = C.polynomial\n",
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" r = f.degree()\n",
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" m = C.exponent\n",
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" \n",
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" self.curve = C\n",
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" g = reduction(C, g)\n",
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" self.function = g\n",
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" \n",
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" def __repr__(self):\n",
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" return str(self.function)\n",
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" \n",
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" def jth_component(self, j):\n",
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" g = self.function\n",
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" R.<x, y> = PolynomialRing(GF(p), 2)\n",
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" g = R(g)\n",
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" return g.coefficient(y^j)\n",
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" \n",
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" def __add__(self, other):\n",
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" C = self.curve\n",
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" g1 = self.function\n",
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" g2 = other.function\n",
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" g = reduction(C, g1 + g2)\n",
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" return superelliptic_function(C, g)\n",
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" \n",
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" def __sub__(self, other):\n",
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" C = self.curve\n",
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" g1 = self.function\n",
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" g2 = other.function\n",
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" g = reduction(C, g1 - g2)\n",
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" return superelliptic_function(C, g)\n",
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" \n",
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" def __mul__(self, other):\n",
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" C = self.curve\n",
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" g1 = self.function\n",
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" g2 = other.function\n",
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" g = reduction(C, g1 * g2)\n",
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" return superelliptic_function(C, g)\n",
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" \n",
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" def __truediv__(self, other):\n",
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" C = self.curve\n",
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" g1 = self.function\n",
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" g2 = other.function\n",
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" g = reduction(C, g1 / g2)\n",
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" return superelliptic_function(C, g)\n",
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" \n",
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"def diffn(self):\n",
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" C = self.curve\n",
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" f = C.polynomial\n",
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" m = C.exponent\n",
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" g = self.function\n",
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" A = g.derivative(x)\n",
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" B = g.derivative(y)*f.derivative(x)/(m*y^(m-1))\n",
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" return superelliptic_form(C, A+B)\n",
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" \n",
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"class superelliptic_form:\n",
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" def __init__(self, C, g):\n",
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" R.<x, y> = PolynomialRing(GF(p), 2)\n",
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" RR = FractionField(R)\n",
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" g = RR(reduction_form(C, g))\n",
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" self.form = g\n",
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" self.curve = C \n",
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" \n",
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" def __add__(self, other):\n",
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" C = self.curve\n",
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" g1 = self.form\n",
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" g2 = other.form\n",
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" g = reduction(C, g1 + g2)\n",
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" return superelliptic_form(C, g)\n",
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" \n",
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" def __sub__(self, other):\n",
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" C = self.curve\n",
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" g1 = self.form\n",
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" g2 = other.form\n",
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" g = reduction(C, g1 - g2)\n",
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" return superelliptic_form(C, g)\n",
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" \n",
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" def __repr__(self):\n",
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" g = self.form\n",
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" if len(str(g)) == 1:\n",
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" return str(g) + ' dx'\n",
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" return '('+str(g) + ') dx'\n",
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" \n",
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" def jth_component(self, j):\n",
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" g = self.form\n",
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" R.<x, y> = PolynomialRing(GF(p), 2)\n",
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" g = R(g)\n",
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" return g.coefficient(y^j)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 220,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"{0: (1/y) dx}"
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]
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},
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"execution_count": 220,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"C = superelliptic(x^3 + x + 2, 2, 5)\n",
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"C.basis_holomorphic_differentials()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 179,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"0"
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]
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},
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"execution_count": 179,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"A.degree(y)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 180,
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"metadata": {},
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"outputs": [],
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"source": [
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"p = 5\n",
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"R.<x, y> = PolynomialRing(GF(p), 2)\n",
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"g = x^6*y^2 + y^2"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 181,
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"metadata": {},
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"outputs": [],
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"source": [
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"omega = diffn(superelliptic_function(C, y^2))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 183,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"-2*x^2 + 1"
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]
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},
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"execution_count": 183,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"omega.jth_component(0)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 33,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"y"
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]
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},
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"execution_count": 33,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"R.<x, y> = PolynomialRing(GF(p), 2)\n",
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"g1 = x^3*y^7 + x^2*y^9\n",
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"g2 = x^2*y + y^6\n",
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"R1.<x> = PolynomialRing(GF(p))\n",
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"R2 = FractionField(R1)\n",
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"R3.<y> = PolynomialRing(R2)\n",
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"\n",
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"xgcd(R3(g1), R3(g2))[1]*R3(g1) + xgcd(R3(g1), R3(g2))[2]*R3(g2)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 39,
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"metadata": {},
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"outputs": [],
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"source": [
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"H = HyperellipticCurve(x^5 - x + 1)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 40,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"Hyperelliptic Curve over Finite Field of size 5 defined by y^2 = x^5 + 4*x + 1"
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]
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},
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"execution_count": 40,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"H"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 84,
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"metadata": {},
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"outputs": [],
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"source": [
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"f = x^3 + x + 2"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 86,
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"metadata": {},
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||
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"outputs": [
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||
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{
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||
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"data": {
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||
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"text/plain": [
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"-2*x^2 + 1"
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]
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||
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},
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||
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"execution_count": 86,
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||
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"metadata": {},
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||
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"output_type": "execute_result"
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}
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],
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"source": [
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"f.derivative(x)"
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]
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},
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{
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"cell_type": "code",
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||
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"execution_count": 213,
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||
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"metadata": {},
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||
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"outputs": [],
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||
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"source": [
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"R1.<x> = PolynomialRing(GF(p))\n",
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||
|
|
"R2 = FractionField(R1)\n",
|
||
|
|
"R3.<y> = PolynomialRing(R2)\n",
|
||
|
|
"g = y^2/x + y/(x+1) "
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 218,
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/plain": [
|
||
|
|
"[0, 1/(x + 1), 1/x]"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 218,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"g.coefficients(sparse = false)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"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
|
||
|
|
}
|