From 5f012a164ef7c4b643ebc86fe0d953284bbc2339 Mon Sep 17 00:00:00 2001
From: jgarnek <jgarnek@amu.edu.pl>
Date: Mon, 23 Aug 2021 14:55:37 +0200
Subject: [PATCH] macierz froba prawie dziala; problem z najstarszym wspol

---
 deRhamComputation.ipynb |  34 +++-
 superelliptic.ipynb     | 353 ++++++++++++++++++++++++++++++++++------
 2 files changed, 336 insertions(+), 51 deletions(-)

diff --git a/deRhamComputation.ipynb b/deRhamComputation.ipynb
index 77ad051..c9c250d 100644
--- a/deRhamComputation.ipynb
+++ b/deRhamComputation.ipynb
@@ -22,7 +22,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 169,
+   "execution_count": 2,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -46,7 +46,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 170,
+   "execution_count": 3,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -79,7 +79,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 208,
+   "execution_count": 4,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -213,7 +213,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 209,
+   "execution_count": 5,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -261,7 +261,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 242,
+   "execution_count": 6,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -456,6 +456,30 @@
     "baza_dr(m, f, 0, p)"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "{0: [1, 0, 3], 1: [0, 2/x, 3]}"
+      ]
+     },
+     "execution_count": 8,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "p = 5\n",
+    "R.<x> = PolynomialRing(GF(p))\n",
+    "f = x^3 + x + 2\n",
+    "m = 7\n",
+    "baza_dr(m, f, 3, p)"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": null,
diff --git a/superelliptic.ipynb b/superelliptic.ipynb
index 1d92c3b..cf5fcb9 100644
--- a/superelliptic.ipynb
+++ b/superelliptic.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 115,
+   "execution_count": 118,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -75,7 +75,7 @@
     "                    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^j/x^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",
@@ -92,6 +92,27 @@
     "    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",
@@ -238,6 +259,27 @@
     "            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.<x> = PolynomialRing(GF(p))\n",
+    "        Fx = FractionField(Rx)\n",
+    "        FxRy.<y> = 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",
@@ -276,6 +318,7 @@
     "            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",
@@ -306,13 +349,22 @@
     "    def __repr__(self):\n",
     "        return \"(\" + str(self.omega0) + \", \" + str(self.f) + \", \" + str(self.omega8) + \")\" \n",
     "    \n",
-    "    def basis_coeffs(self):\n",
+    "    def verschiebung(self):\n",
     "        C = self.curve\n",
-    "        g = self.f\n",
-    "        basis = C.basis_de_rham()\n",
+    "        omega = self.omega0\n",
+    "        p = C.characteristic\n",
+    "        Rx.<x> = 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.<x> = 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)\n",
+    "        print(self, self.is_cocycle())\n",
     "        C = self.curve\n",
     "        p = C.characteristic\n",
     "        m = C.exponent\n",
@@ -329,41 +381,63 @@
     "        omega = self.omega0\n",
     "        fct = self.f\n",
     "        \n",
-    "        if fct.function == Rx(0) and omega.form == Rx(0):\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 = d = degree_of_rational_fctn(omega_j)\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",
-    "                    elt =- superelliptic_cech(C, elt.omega0, elt.f - FxRy(y^j*x^d)).mult(a)\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 =-superelliptic_cech(diffn(G), G).mult(a)\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.<x> = PolynomialRing(GF(p))\n",
@@ -376,9 +450,12 @@
     "    return(d1 - d2)\n",
     "\n",
     "def coeff_of_rational_fctn(f):\n",
+    "    print('coeff', f)\n",
     "    Rx.<x> = 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",
@@ -396,58 +473,192 @@
     "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))"
+    "    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.<x> = 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": 116,
+   "execution_count": 119,
    "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)}"
-      ]
-     },
-     "execution_count": 116,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
+   "outputs": [],
    "source": [
     "p = 5\n",
     "C = superelliptic(x^3 + x + 2, 7, p)\n",
-    "C.basis_de_rham()\n",
+    "baza = C.basis_de_rham()\n",
     "#C.basis_holomorphic_differentials()"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 117,
+   "execution_count": 120,
    "metadata": {},
-   "outputs": [],
+   "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": [
-    "RxRy.<x, y> = PolynomialRing(GF(p), 2)\n",
-    "w1 = superelliptic_cech(C, superelliptic_form(C, (1/y^5)), superelliptic_function(C, 0))\n",
-    "w2 = superelliptic_cech(C, superelliptic_form(C,2*x/y^4), superelliptic_function(C, 2/x*y^4))\n",
-    "w = w1+w2+w2"
+    "for w in baza.values():\n",
+    "    print(w, w.is_cocycle())"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 118,
+   "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<ipython-input-121-e1ff68010d76>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mC\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfrobenius_matrix\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
+      "\u001b[0;32m<ipython-input-118-ac653a450b02>\u001b[0m in \u001b[0;36mfrobenius_matrix\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    103\u001b[0m         \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mw\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mbasis\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mitems\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    104\u001b[0m             \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'w'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mw\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 105\u001b[0;31m             \u001b[0mv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mw\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfrobenius\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcoordinates\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    106\u001b[0m             \u001b[0mM\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m:\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    107\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mM\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m<ipython-input-118-ac653a450b02>\u001b[0m in \u001b[0;36mcoordinates\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    414\u001b[0m                     \u001b[0ma\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcoeff_of_rational_fctn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfct_j\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    415\u001b[0m                     \u001b[0melt\u001b[0m \u001b[0;34m=\u001b[0m\u001b[0mself\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0msuperelliptic_cech\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdiffn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmult\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 416\u001b[0;31m                     \u001b[0;32mreturn\u001b[0m \u001b[0melt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcoordinates\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    417\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    418\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mvector\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mg\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m<ipython-input-118-ac653a450b02>\u001b[0m in \u001b[0;36mcoordinates\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    414\u001b[0m                     \u001b[0ma\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcoeff_of_rational_fctn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfct_j\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    415\u001b[0m                     \u001b[0melt\u001b[0m \u001b[0;34m=\u001b[0m\u001b[0mself\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0msuperelliptic_cech\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdiffn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmult\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 416\u001b[0;31m                     \u001b[0;32mreturn\u001b[0m \u001b[0melt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcoordinates\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    417\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    418\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mvector\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mg\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m<ipython-input-118-ac653a450b02>\u001b[0m in \u001b[0;36mcoordinates\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    414\u001b[0m                     \u001b[0ma\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcoeff_of_rational_fctn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfct_j\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    415\u001b[0m                     \u001b[0melt\u001b[0m \u001b[0;34m=\u001b[0m\u001b[0mself\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0msuperelliptic_cech\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdiffn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmult\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 416\u001b[0;31m                     \u001b[0;32mreturn\u001b[0m \u001b[0melt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcoordinates\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    417\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    418\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mvector\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mg\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m<ipython-input-118-ac653a450b02>\u001b[0m in \u001b[0;36mcoordinates\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    414\u001b[0m                     \u001b[0ma\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcoeff_of_rational_fctn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfct_j\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    415\u001b[0m                     \u001b[0melt\u001b[0m \u001b[0;34m=\u001b[0m\u001b[0mself\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0msuperelliptic_cech\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdiffn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmult\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 416\u001b[0;31m                     \u001b[0;32mreturn\u001b[0m \u001b[0melt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcoordinates\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    417\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    418\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mvector\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mg\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m<ipython-input-118-ac653a450b02>\u001b[0m in \u001b[0;36mcoordinates\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    414\u001b[0m                     \u001b[0ma\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcoeff_of_rational_fctn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfct_j\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    415\u001b[0m                     \u001b[0melt\u001b[0m \u001b[0;34m=\u001b[0m\u001b[0mself\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0msuperelliptic_cech\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdiffn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmult\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 416\u001b[0;31m                     \u001b[0;32mreturn\u001b[0m \u001b[0melt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcoordinates\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    417\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    418\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mvector\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mg\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m<ipython-input-118-ac653a450b02>\u001b[0m in \u001b[0;36mcoordinates\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    414\u001b[0m                     \u001b[0ma\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcoeff_of_rational_fctn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfct_j\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    415\u001b[0m                     \u001b[0melt\u001b[0m \u001b[0;34m=\u001b[0m\u001b[0mself\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0msuperelliptic_cech\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdiffn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmult\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 416\u001b[0;31m                     \u001b[0;32mreturn\u001b[0m \u001b[0melt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcoordinates\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    417\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    418\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mvector\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mg\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m<ipython-input-118-ac653a450b02>\u001b[0m in \u001b[0;36mcoordinates\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    414\u001b[0m                     \u001b[0ma\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcoeff_of_rational_fctn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfct_j\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    415\u001b[0m                     \u001b[0melt\u001b[0m \u001b[0;34m=\u001b[0m\u001b[0mself\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0msuperelliptic_cech\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdiffn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmult\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 416\u001b[0;31m                     \u001b[0;32mreturn\u001b[0m \u001b[0melt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcoordinates\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    417\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    418\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mvector\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mg\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m<ipython-input-118-ac653a450b02>\u001b[0m in \u001b[0;36mcoordinates\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    399\u001b[0m                     \u001b[0ma1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcoeff_of_rational_fctn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mbasis\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mindex\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mjth_component\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[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    400\u001b[0m                     \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0ma1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mindex\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mbasis\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mindex\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_cocycle\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--> 401\u001b[0;31m                     \u001b[0melt\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mbasis\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mindex\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmult\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m/\u001b[0m\u001b[0ma1\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    402\u001b[0m                     \u001b[0;32mreturn\u001b[0m \u001b[0melt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcoordinates\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m/\u001b[0m\u001b[0ma1\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mvector\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mGF\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mi\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0mindex\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    403\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/element.pyx\u001b[0m in \u001b[0;36msage.structure.element.Element.__truediv__ (build/cythonized/sage/structure/element.c:12837)\u001b[0;34m()\u001b[0m\n\u001b[1;32m   1716\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;34m<\u001b[0m\u001b[0mElement\u001b[0m\u001b[0;34m>\u001b[0m\u001b[0mleft\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_div_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mright\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1717\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mBOTH_ARE_ELEMENT\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcl\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1718\u001b[0;31m             \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[0m\u001b[1;32m   1719\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1720\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/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/coerce.pyx\u001b[0m in \u001b[0;36msage.structure.coerce.CoercionModel.bin_op (build/cythonized/sage/structure/coerce.c:10143)\u001b[0;34m()\u001b[0m\n\u001b[1;32m   1209\u001b[0m             \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_record_exception\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1210\u001b[0m         \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1211\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mPyObject_CallObject\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mop\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mxy\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1212\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1213\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mop\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0mmul\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/element.pyx\u001b[0m in \u001b[0;36msage.structure.element.Element.__truediv__ (build/cythonized/sage/structure/element.c:12802)\u001b[0;34m()\u001b[0m\n\u001b[1;32m   1714\u001b[0m         \u001b[0mcdef\u001b[0m \u001b[0mint\u001b[0m \u001b[0mcl\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mclassify_elements\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mleft\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mright\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1715\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mHAVE_SAME_PARENT\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcl\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1716\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;34m<\u001b[0m\u001b[0mElement\u001b[0m\u001b[0;34m>\u001b[0m\u001b[0mleft\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_div_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mright\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1717\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mBOTH_ARE_ELEMENT\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcl\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1718\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mcoercion_model\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbin_op\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mleft\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mright\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtruediv\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/rings/finite_rings/integer_mod.pyx\u001b[0m in \u001b[0;36msage.rings.finite_rings.integer_mod.IntegerMod_int._div_ (build/cythonized/sage/rings/finite_rings/integer_mod.c:27689)\u001b[0;34m()\u001b[0m\n\u001b[1;32m   2540\u001b[0m             \u001b[0mright_inverse\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__modulus\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0minverses\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m<\u001b[0m\u001b[0mIntegerMod_int\u001b[0m\u001b[0;34m>\u001b[0m\u001b[0mright\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mivalue\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2541\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mright_inverse\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2542\u001b[0;31m                 \u001b[0;32mraise\u001b[0m \u001b[0mZeroDivisionError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34mf\"inverse of Mod({right}, {self.__modulus.sageInteger}) does not exist\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2543\u001b[0m             \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2544\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_new_c\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mivalue\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;34m<\u001b[0m\u001b[0mIntegerMod_int\u001b[0m\u001b[0;34m>\u001b[0m\u001b[0mright_inverse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mivalue\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__modulus\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mint32\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;31mZeroDivisionError\u001b[0m: inverse of Mod(0, 5) does not exist"
+     ]
+    }
+   ],
+   "source": [
+    "C.frobenius_matrix()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
    "metadata": {},
    "outputs": [
     {
@@ -455,16 +666,17 @@
      "output_type": "stream",
      "text": [
       "(((-x*y + 1)/y^5) dx, 4/x*y^4, ((-x^3*y + x*y^2 + x^2 - 2*y^2)/(x^2*y^5)) dx)\n",
-      "((1/y^5) dx, 0, (1/y^5) dx)\n"
+      "((1/y^5) dx, 0, (1/y^5) dx)\n",
+      "(0 dx, 0, 0 dx)\n"
      ]
     },
     {
      "data": {
       "text/plain": [
-       "(0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0)"
+       "(0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0)"
       ]
      },
-     "execution_count": 118,
+     "execution_count": 4,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -850,6 +1062,55 @@
     "degrees1_inv"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 28,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "Superelliptic curve with the equation y^7 = x^3 + x + 2 over finite field with 5 elements."
+      ]
+     },
+     "execution_count": 28,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "C"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 29,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "basis = C.basis_de_rham()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 32,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "dict_items([(0, ((x/y) dx, 2/x*y, ((x^3*y^5 - x^3 + x - 1)/(x^2*y^6)) dx)), (1, (((-1)/y) dx, 2/x^2*y, ((-x^3*y^5 + x^3 - 2*x - 2)/(x^3*y^6)) dx)), (2, (((-2*x)/y^2) dx, 2/x*y^2, ((-2*x^3*y^3 + x^3 - 1)/(x^2*y^5)) dx)), (3, ((1/y^2) dx, 2/x^2*y^2, ((x^3*y^3 - 2*x^3 + 2*x - 2)/(x^3*y^5)) dx)), (4, ((1/y^3) dx, 0, (1/y^3) dx)), (5, (0 dx, 2/x*y^3, ((-2*x^3 - x - 1)/(x^2*y^4)) dx)), (6, ((1/y^4) dx, 0, (1/y^4) dx)), (7, ((2*x/y^4) dx, 2/x*y^4, ((2*x^3 - 2*x*y - y)/(x^2*y^4)) dx)), (8, ((1/y^5) dx, 0, (1/y^5) dx)), (9, ((x/y^5) dx, 0, (x/y^5) dx)), (10, ((1/y^6) dx, 0, (1/y^6) dx)), (11, ((x/y^6) dx, 0, (x/y^6) dx))])"
+      ]
+     },
+     "execution_count": 32,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "basis.items()"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": null,