obliczanie bazy C przy inicjacji
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parent
15c6350fb7
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@ -2,17 +2,56 @@
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 137,
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"execution_count": 206,
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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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" Rx.<x> = PolynomialRing(GF(p))\n",
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" Rxy.<x, y> = PolynomialRing(GF(p), 2)\n",
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" Fxy = FractionField(Rxy)\n",
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" self.polynomial = Rx(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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" r = Rx(f).degree()\n",
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" delta = GCD(r, m)\n",
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" #########basis of holomorphic differentials and de Rham\n",
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" \n",
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" basis_holo = {}\n",
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" basis = {}\n",
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" degrees0 = {}\n",
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" degrees1 = {}\n",
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" k = 0\n",
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" \n",
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" for j in range(1, m):\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_holo[k] = superelliptic_form(self, Fxy(x^(i-1)/y^j))\n",
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" basis[k] = superelliptic_cech(self, basis_holo[k], superelliptic_function(self, Rx(0))) \n",
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" degrees0[k] = (i-1, j)\n",
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" k = k+1\n",
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" self.basis_holomorphic_differentials=basis_holo\n",
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" ## non-holomorphic elts of H^1_dR\n",
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" t = len(basis)\n",
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" \n",
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" for j in range(1, m):\n",
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" for i in range(1, r):\n",
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" if (r*(m-j) - m*i >= delta):\n",
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" s = Rx(m-j)*Rx(x)*Rx(f.derivative()) - Rx(m)*Rx(i)*f\n",
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" psi = Rx(cut(s, i))\n",
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" basis[t] = superelliptic_cech(self, superelliptic_form(self, Fxy(psi/y^j)), superelliptic_function(self, Fxy(m*y^(m-j)/x^i)))\n",
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" if psi != Rx(0):\n",
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" degrees0[t] = (psi.degree(), j)\n",
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" else:\n",
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" degrees0[t] = (0, j)\n",
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" degrees1[t] = (-i, m-j)\n",
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" t += 1\n",
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" self.degree_de_rham0 = degrees0\n",
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" self.degree_de_rham1 = degrees1\n",
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" self.basis_de_rham = basis\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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@ -31,70 +70,9 @@
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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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" Rxy.<x, y> = PolynomialRing(GF(p), 2)\n",
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" Fxy = FractionField(Rxy)\n",
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" \n",
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" basis = {}\n",
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" k = 0\n",
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" if j == 'all':\n",
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" for j in range(1, m):\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(self, Fxy(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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" 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(self, Fxy(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 degree_and_basis_de_rham(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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" Rx.<x> = PolynomialRing(GF(p))\n",
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" Rxy.<x, y> = PolynomialRing(GF(p), 2)\n",
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" Fxy = FractionField(Rxy)\n",
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" basis = {}\n",
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" degrees0 = {}\n",
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" degrees1 = {}\n",
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" t = 0\n",
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" if j == 'all':\n",
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" for j in range(1, m):\n",
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" holo = C.basis_holomorphic_differentials(j)\n",
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" for k in range(0, len(holo)):\n",
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" basis[t] = superelliptic_cech(self, holo[k], superelliptic_function(self, Rx(0))) \n",
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" g = Rx(holo[k].jth_component(j))\n",
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" degrees0[t] = (g.degree(), j)\n",
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" t += 1\n",
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" for i in range(1, r):\n",
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" if (r*(m-j) - m*i >= delta):\n",
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" s = Rx(m-j)*Rx(x)*Rx(f.derivative()) - Rx(m)*Rx(i)*f\n",
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" psi = Rx(cut(s, i))\n",
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" basis[t] = superelliptic_cech(self, superelliptic_form(self, Fxy(psi/y^j)), superelliptic_function(self, Fxy(m*y^(m-j)/x^i)))\n",
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" degrees0[t] = (psi.degree(), j)\n",
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" degrees1[t] = (-i, m-j)\n",
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" t += 1\n",
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" return basis, degrees0, degrees1\n",
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" \n",
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" def degree_de_rham(self, i, j='all'):\n",
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" basis, degrees0, degrees1 = self.degree_and_basis_de_rham(j)\n",
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" if i==0:\n",
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" return degrees0\n",
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" \n",
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" if i==1:\n",
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" return degrees1\n",
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" \n",
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" def basis_de_rham(self, j = 'all'): \n",
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" basis, degrees0, degrees1 = self.degree_and_basis_de_rham(j)\n",
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" return basis\n",
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" \n",
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" def verschiebung_matrix(self):\n",
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" basis = self.basis_de_rham()\n",
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" basis = self.basis_de_rham\n",
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" g = self.genus()\n",
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" p = self.characteristic\n",
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" M = matrix(GF(p), 2*g, 2*g)\n",
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@ -104,7 +82,7 @@
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" return M\n",
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" \n",
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" def frobenius_matrix(self):\n",
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" basis = self.basis_de_rham()\n",
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" basis = self.basis_de_rham\n",
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" g = self.genus()\n",
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" p = self.characteristic\n",
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" M = matrix(GF(p), 2*g, 2*g)\n",
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@ -369,11 +347,11 @@
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" Fx = FractionField(Rx)\n",
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" FxRy.<y> = PolynomialRing(Fx)\n",
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" g = C.genus()\n",
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" degrees0 = C.degree_de_rham(0)\n",
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" degrees0 = C.degree_de_rham0\n",
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" degrees0_inv = {b:a for a, b in degrees0.items()} \n",
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" degrees1 = C.degree_de_rham(1)\n",
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" degrees1 = C.degree_de_rham1\n",
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" degrees1_inv = {b:a for a, b in degrees1.items()}\n",
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" basis = C.basis_de_rham()\n",
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" basis = C.basis_de_rham\n",
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" \n",
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" omega = self.omega0\n",
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" fct = self.f\n",
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@ -476,71 +454,134 @@
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},
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{
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"cell_type": "code",
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"execution_count": 146,
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"execution_count": 207,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"1\n"
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"4\n",
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"2\n"
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]
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}
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],
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"source": [
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"p = 5\n",
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"C = superelliptic(x^3 + x + 2, 2, p)\n",
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"baza = C.basis_de_rham()\n",
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"p = 7\n",
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"Rx.<x> = PolynomialRing(GF(p))\n",
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"C = superelliptic(x^3 + x + 3, 5, p)\n",
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"baza = C.basis_de_rham\n",
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"print(C.genus())\n",
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"#E = EllipticCurve(GF(p), [1, 2])\n",
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"print(E.trace_of_frobenius())\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": 147,
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"metadata": {
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"scrolled": false
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},
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"outputs": [],
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"source": [
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"A = C.frobenius_matrix()\n",
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"B = C.verschiebung_matrix()"
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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": 148,
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"execution_count": 208,
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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 0]\n",
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"[1 2]"
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"{0: (-1, 1), 1: (0, 1), 2: (1, 2), 3: (1, 3)}"
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]
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},
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"execution_count": 148,
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"execution_count": 208,
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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"
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"C.degree_de_rham0"
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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": 149,
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"execution_count": 209,
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"metadata": {
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"scrolled": false
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},
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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: (0 dx, 5/x*y^4, ((x + 1)/(x^2*y)) dx),\n",
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" 1: ((2/y) dx, 5/x^2*y^4, ((-x + 2)/(x^3*y)) dx),\n",
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" 2: (((-3*x)/y^2) dx, 5/x*y^3, ((2*x + 1)/(x^2*y^2)) dx),\n",
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" 3: ((x/y^3) dx, 5/x*y^2, ((3*x + 1)/(x^2*y^3)) dx)}"
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]
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},
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"execution_count": 209,
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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.basis_de_rham"
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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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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"False\n",
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"False\n",
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"False\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n",
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"True\n"
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]
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}
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],
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"source": [
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"for i in range(0, 30):\n",
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" print((B^i).image() == (B^(i+1)).image())"
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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": 157,
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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 0]\n",
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"[4 0]"
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"[7 0]\n",
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"[6 0]"
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]
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},
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"execution_count": 149,
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"execution_count": 157,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -554,20 +595,25 @@
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"execution_count": 150,
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"metadata": {},
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"outputs": [],
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"source": [
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"E = EllipticCurve(GF(p), [1, 2])"
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]
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"execution_count": 151,
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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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"outputs": [
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{
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"data": {
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"text/plain": [
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"2"
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]
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},
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"execution_count": 151,
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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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},
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{
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"cell_type": "code",
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