diff --git a/superelliptic.ipynb b/superelliptic.ipynb
index f56a2ac..6af5b96 100644
--- a/superelliptic.ipynb
+++ b/superelliptic.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 41,
+   "execution_count": 67,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -157,6 +157,15 @@
     "            M[i, :] = v\n",
     "        return M     \n",
     "    \n",
+    "    def p_rank(self):\n",
+    "        return self.cartier_matrix().rank()\n",
+    "    \n",
+    "    def final_type(self):\n",
+    "        F = self.frobenius_matrix()\n",
+    "        V = self.verschiebung_matrix()\n",
+    "        p = self.characteristic\n",
+    "        return flag(F, V, p)\n",
+    "    \n",
     "def reduction(C, g):\n",
     "    p = C.characteristic\n",
     "    Rxy.<x, y> = PolynomialRing(GF(p), 2)\n",
@@ -305,12 +314,7 @@
     "    def __rmul__(self, constant):\n",
     "        C = self.curve\n",
     "        omega = self.form\n",
-    "        return superelliptic_form(C, constant*omega)     \n",
-    "    \n",
-    "    def mult(self, constant):\n",
-    "        C = self.curve\n",
-    "        omega = self.form\n",
-    "        return superelliptic_form(C, constant*omega)    \n",
+    "        return superelliptic_form(C, constant*omega)        \n",
     "    \n",
     "    def cartier(self):\n",
     "        C = self.curve\n",
@@ -352,7 +356,7 @@
     "                index = degrees_holo_inv[(d, j)]\n",
     "                a = coeff_of_rational_fctn(omega_j, p)\n",
     "                a1 = coeff_of_rational_fctn(basis[index].jth_component(j), p)\n",
-    "                elt = self - basis[index].mult(a/a1)\n",
+    "                elt = self - (a/a1)*basis[index]\n",
     "                return elt.coordinates() + a/a1*vector([GF(p)(i == index) for i in range(0, g)])\n",
     "            \n",
     "        return vector(g*[0])\n",
@@ -413,14 +417,14 @@
     "    def __sub__(self, other):\n",
     "        C = self.curve\n",
     "        return superelliptic_cech(C, self.omega0 - other.omega0, self.f - other.f)\n",
-    "    \n",
-    "    def mult(self, constant):\n",
+    "\n",
+    "    def __rmul__(self, constant):\n",
     "        C = self.curve\n",
     "        w1 = self.omega0.form\n",
     "        f1 = self.f.function\n",
     "        w2 = superelliptic_form(C, constant*w1)\n",
     "        f2 = superelliptic_function(C, constant*f1)\n",
-    "        return superelliptic_cech(C, w2, f2)\n",
+    "        return superelliptic_cech(C, w2, f2)    \n",
     "    \n",
     "    def __repr__(self):\n",
     "        return \"(\" + str(self.omega0) + \", \" + str(self.f) + \", \" + str(self.omega8) + \")\" \n",
@@ -466,7 +470,7 @@
     "                    index = degrees_holo_inv[(d, j)]\n",
     "                    a = coeff_of_rational_fctn(omega_j, p)\n",
     "                    a1 = coeff_of_rational_fctn(basis[index].omega0.jth_component(j), p)\n",
-    "                    elt = self - basis[index].mult(a/a1)\n",
+    "                    elt = self - (a/a1)*basis[index]\n",
     "                    return elt.coordinates() + a/a1*vector([GF(p)(i == index) for i in range(0, 2*g)])\n",
     "                    \n",
     "        for j in range(1, m):\n",
@@ -477,7 +481,7 @@
     "                if (d, j) in degrees1.values():\n",
     "                    index = degrees1_inv[(d, j)]\n",
     "                    a = coeff_of_rational_fctn(fct_j, p)\n",
-    "                    elt = self - basis[index].mult(a/m)\n",
+    "                    elt = self - (a/m)*basis[index]\n",
     "                    return elt.coordinates() + a/m*vector([GF(p)(i == index) for i in range(0, 2*g)])\n",
     "                \n",
     "                if d<0:\n",
@@ -489,7 +493,7 @@
     "                if (fct_j != Rx(0)):\n",
     "                    G = superelliptic_function(C, y^j*x^d)\n",
     "                    a = coeff_of_rational_fctn(fct_j, p)\n",
-    "                    elt =self - superelliptic_cech(C, diffn(G), G).mult(a)\n",
+    "                    elt =self - a*superelliptic_cech(C, diffn(G), G)\n",
     "                    return elt.coordinates()\n",
     "\n",
     "        return vector(2*g*[0])\n",
@@ -556,7 +560,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 31,
+   "execution_count": 68,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -609,14 +613,11 @@
     "        if final_type[i] == -1 and final_type[dim - i] != -1:\n",
     "            i1 = dim - i\n",
     "            final_type[i] = final_type[i1] - i1 + dim/2\n",
-    "    print('test', final_type)\n",
     "    \n",
     "    final_type[0] = 0\n",
     "    for i in range(1, dim+1):\n",
     "        if final_type[i] == -1:\n",
     "            final_type[i] = min(final_type[i-1] + 1, dim/2)\n",
-    "            \n",
-    "    print('poprawiony ft', final_type)\n",
     "    \n",
     "    if is_final(final_type, dim/2):\n",
     "        return final_type[1:dim/2 + 1]\n",
@@ -638,60 +639,1423 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 18,
+   "execution_count": null,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "0 3\n",
-      "test [0, -1, -1, -1, 4, -1, -1, -1, 4]\n",
-      "poprawiony ft [0, 1, 2, 3, 4, 4, 4, 4, 4]\n",
-      "0 3 4 [1, 2, 3, 4]\n",
-      "0 4\n",
-      "test [0, -1, -1, -1, 4, -1, -1, -1, 4]\n",
-      "poprawiony ft [0, 1, 2, 3, 4, 4, 4, 4, 4]\n",
-      "0 4 4 [1, 2, 3, 4]\n",
-      "1 1\n",
-      "test [0, -1, -1, -1, 4, -1, -1, -1, 4]\n",
-      "poprawiony ft [0, 1, 2, 3, 4, 4, 4, 4, 4]\n",
-      "1 1 4 [1, 2, 3, 4]\n",
-      "1 2\n"
+      "x^9 + x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 8*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 8*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 8*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 8*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 8*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 8*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 8*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 8*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 8*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 8*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + x^8 + 8*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 2*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 2*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 2*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 2*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 2*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 2*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 2*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 2*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 2*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 2*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 2*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 2*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 2*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 2*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 2*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 2*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 2*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 2*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 2*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 2*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 2*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 2*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 2*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 7*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 3*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 7*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 3*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 7*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 3*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 7*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 3*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 7*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 3*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 7*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 3*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 4*x 3\n"
      ]
     },
     {
-     "ename": "KeyboardInterrupt",
-     "evalue": "",
-     "output_type": "error",
-     "traceback": [
-      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
-      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
-      "\u001b[0;32m<ipython-input-18-afeec6e02e63>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      6\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_smooth\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      7\u001b[0m             \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 8\u001b[0;31m             \u001b[0mV\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mverschiebung_matrix\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      9\u001b[0m             \u001b[0mF\u001b[0m \u001b[0;34m=\u001b[0m \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[1;32m     10\u001b[0m             \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mV\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrank\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mflag\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mF\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mV\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mp\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-17-0d6574174a68>\u001b[0m in \u001b[0;36mverschiebung_matrix\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    123\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[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mbasis\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    124\u001b[0m             \u001b[0mw\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mbasis\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[0m\n\u001b[0;32m--> 125\u001b[0;31m             \u001b[0mv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mw\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mverschiebung\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    126\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    127\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-17-0d6574174a68>\u001b[0m in \u001b[0;36mcoordinates\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    450\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[0momega0\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[0mp\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    451\u001b[0m                     \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[0;32m--> 452\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[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[0m\u001b[1;32m    453\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    454\u001b[0m         \u001b[0;32mfor\u001b[0m \u001b[0mj\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;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mm\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",
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-      "\u001b[0;32m<ipython-input-17-0d6574174a68>\u001b[0m in \u001b[0;36mcoeff_of_rational_fctn\u001b[0;34m(f, p)\u001b[0m\n\u001b[1;32m    502\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    503\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mcoeff_of_rational_fctn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mp\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--> 504\u001b[0;31m     \u001b[0mRx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mPolynomialRing\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[0mnames\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'x'\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[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mRx\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_first_ngens\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\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    505\u001b[0m     \u001b[0mFx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mFractionField\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mRx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    506\u001b[0m     \u001b[0mf\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mFx\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;32m/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/rings/polynomial/polynomial_ring_constructor.py\u001b[0m in \u001b[0;36mPolynomialRing\u001b[0;34m(base_ring, *args, **kwds)\u001b[0m\n\u001b[1;32m    631\u001b[0m             \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"you must specify the names of the variables\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    632\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 633\u001b[0;31m     \u001b[0mnames\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnormalize_names\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnames\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    634\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    635\u001b[0m     \u001b[0;31m# At this point, we have only handled the \"names\" keyword if it was\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/category_object.pyx\u001b[0m in \u001b[0;36msage.structure.category_object.normalize_names (build/cythonized/sage/structure/category_object.c:8424)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    922\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    923\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 924\u001b[0;31m \u001b[0mcpdef\u001b[0m \u001b[0mnormalize_names\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mPy_ssize_t\u001b[0m \u001b[0mngens\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnames\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    925\u001b[0m     r\"\"\"\n\u001b[1;32m    926\u001b[0m     \u001b[0mReturn\u001b[0m \u001b[0ma\u001b[0m \u001b[0mtuple\u001b[0m \u001b[0mof\u001b[0m \u001b[0mstrings\u001b[0m \u001b[0mof\u001b[0m \u001b[0mvariable\u001b[0m \u001b[0mnames\u001b[0m \u001b[0mof\u001b[0m \u001b[0mlength\u001b[0m \u001b[0mngens\u001b[0m \u001b[0mgiven\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/category_object.pyx\u001b[0m in \u001b[0;36msage.structure.category_object.normalize_names (build/cythonized/sage/structure/category_object.c:7828)\u001b[0;34m()\u001b[0m\n\u001b[1;32m   1019\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnames\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mtuple\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlist\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   1020\u001b[0m         \u001b[0;31m# Convert names to strings and strip whitespace\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1021\u001b[0;31m         \u001b[0mnames\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mstrip\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mx\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mnames\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   1022\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   1023\u001b[0m         \u001b[0;31m# Interpret names as string and convert to tuple of strings\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;32msrc/cysignals/signals.pyx\u001b[0m in \u001b[0;36mcysignals.signals.python_check_interrupt\u001b[0;34m()\u001b[0m\n",
-      "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 7*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 3*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 7*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 3*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 7*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 3*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 7*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 3*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 7*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 3*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 3*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 6*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 4*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 6*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 4*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 6*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 4*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 6*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 4*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 6*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 4*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 6*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 4*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 6*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 4*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 6*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 4*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 6*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 4*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 6*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 4*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 6*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 4*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 4*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 10*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 5*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 10*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 5*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 10*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 5*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 10*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 5*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 10*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 5*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 10*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 5*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 10*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 5*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 10*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 5*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 10*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 5*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 10*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 5*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 5*x^8 + 10*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 6*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 10*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 6*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n"
+     ]
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "x^9 + 6*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 10*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 6*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 10*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 6*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 10*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 6*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 10*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 6*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 10*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 6*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 10*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 6*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 10*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 6*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 10*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 6*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 10*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 6*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 2*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 6*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 6*x^8 + 10*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 7*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 6*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 7*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 6*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 7*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 6*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 7*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 6*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 7*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 6*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 7*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 6*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 7*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 6*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 7*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 6*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 7*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 6*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 7*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 6*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 7*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 9*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 10*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 3*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 4*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 5*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 6*x 3\n",
+      "[0, 1, 2, 3]\n",
+      "x^9 + 7*x^8 + 7*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 8*x 3\n",
+      "[1, 2, 2, 3]\n",
+      "x^9 + 7*x^8 + 9*x 3\n"
      ]
     }
    ],
    "source": [
-    "p = 5\n",
+    "p = 11\n",
     "for a in range(0, p):\n",
     "    for b in range(0, p):\n",
-    "        Rx.<x> = PolynomialRing(GF(p))\n",
-    "        C = superelliptic(x^9+a*x^8+b*x^3+x^2+x+1, 2, p)\n",
-    "        if C.is_smooth():\n",
-    "            print(a, b)\n",
-    "            V = C.verschiebung_matrix()\n",
-    "            F = C.frobenius_matrix()\n",
-    "            print(a, b, V.rank(), flag(F, V, p))"
+    "        for c in range(0, p):\n",
+    "            Rx.<x> = PolynomialRing(GF(p))\n",
+    "            C = superelliptic(x^9+a*x^8+c*x, 2, p)\n",
+    "            if C.is_smooth() and C.p_rank() == 3:\n",
+    "                print(C.polynomial, C.p_rank())\n",
+    "                V = C.verschiebung_matrix()\n",
+    "                F = C.frobenius_matrix()\n",
+    "                print(flag(F, V, p))"
    ]
   },
   {
@@ -716,51 +2080,82 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 42,
+   "execution_count": 81,
    "metadata": {
     "scrolled": true
    },
-   "outputs": [],
-   "source": [
-    "Rx.<x> = PolynomialRing(GF(5))\n",
-    "C = superelliptic(x^9+3*x^3+x^2+x+1, 2, 5)\n",
-    "#C.cartier_matrix()"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 38,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "om = C.basis_holomorphic_differentials[0]"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 40,
-   "metadata": {},
    "outputs": [
     {
-     "ename": "AttributeError",
-     "evalue": "'sage.rings.integer.Integer' object has no attribute 'curve'",
-     "output_type": "error",
-     "traceback": [
-      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
-      "\u001b[0;31mAttributeError\u001b[0m                            Traceback (most recent call last)",
-      "\u001b[0;32m<ipython-input-40-15ffd216e2f8>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \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[0mom\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
-      "\u001b[0;32m/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/rings/integer.pyx\u001b[0m in \u001b[0;36msage.rings.integer.Integer.__mul__ (build/cythonized/sage/rings/integer.c:13835)\u001b[0;34m()\u001b[0m\n\u001b[1;32m   1998\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0my\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1999\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2000\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[0moperator\u001b[0m\u001b[0;34m.\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[0m\u001b[1;32m   2001\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2002\u001b[0m     \u001b[0mcpdef\u001b[0m \u001b[0m_mul_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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[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:11104)\u001b[0;34m()\u001b[0m\n\u001b[1;32m   1247\u001b[0m             \u001b[0mmul_method\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mgetattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'__r%s__'\u001b[0m\u001b[0;34m%\u001b[0m\u001b[0mop_name\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1248\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mmul_method\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1249\u001b[0;31m                 \u001b[0mres\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mmul_method\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1250\u001b[0m                 \u001b[0;32mif\u001b[0m \u001b[0mres\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mres\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mNotImplemented\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1251\u001b[0m                     \u001b[0;32mreturn\u001b[0m \u001b[0mres\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;32m<ipython-input-36-ef3705b062c6>\u001b[0m in \u001b[0;36m__rmul__\u001b[0;34m(constant, self)\u001b[0m\n\u001b[1;32m    296\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    297\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m__rmul__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mconstant\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 298\u001b[0;31m         \u001b[0mC\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcurve\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    299\u001b[0m         \u001b[0momega\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mform\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    300\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0msuperelliptic_form\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mconstant\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0momega\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;32m/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/element.pyx\u001b[0m in \u001b[0;36msage.structure.element.Element.__getattr__ (build/cythonized/sage/structure/element.c:4614)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    485\u001b[0m             \u001b[0mAttributeError\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0;34m'LeftZeroSemigroup_with_category.element_class'\u001b[0m \u001b[0mobject\u001b[0m \u001b[0mhas\u001b[0m \u001b[0mno\u001b[0m \u001b[0mattribute\u001b[0m \u001b[0;34m'blah_blah'\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    486\u001b[0m         \"\"\"\n\u001b[0;32m--> 487\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mgetattr_from_category\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mname\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    488\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    489\u001b[0m     \u001b[0mcdef\u001b[0m \u001b[0mgetattr_from_category\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mname\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;32m/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/element.pyx\u001b[0m in \u001b[0;36msage.structure.element.Element.getattr_from_category (build/cythonized/sage/structure/element.c:4723)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    498\u001b[0m         \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    499\u001b[0m             \u001b[0mcls\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mP\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_abstract_element_class\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 500\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mgetattr_from_other_class\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mname\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    501\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    502\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m__dir__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;32m/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/cpython/getattr.pyx\u001b[0m in \u001b[0;36msage.cpython.getattr.getattr_from_other_class (build/cythonized/sage/cpython/getattr.c:2614)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    392\u001b[0m         \u001b[0mdummy_error_message\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcls\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    393\u001b[0m         \u001b[0mdummy_error_message\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mname\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mname\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 394\u001b[0;31m         \u001b[0;32mraise\u001b[0m \u001b[0mAttributeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdummy_error_message\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    395\u001b[0m     \u001b[0mattribute\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m<\u001b[0m\u001b[0mobject\u001b[0m\u001b[0;34m>\u001b[0m\u001b[0mattr\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    396\u001b[0m     \u001b[0;31m# Check for a descriptor (__get__ in Python)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;31mAttributeError\u001b[0m: 'sage.rings.integer.Integer' object has no attribute 'curve'"
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "1\n",
+      "[1]\n"
      ]
     }
    ],
    "source": [
-    "2*om"
+    "p = 5\n",
+    "Rx.<x> = PolynomialRing(GF(p))\n",
+    "C = superelliptic(x^3 - x, 2, p)\n",
+    "if (C.is_smooth()):\n",
+    "    print(C.p_rank())\n",
+    "    print(C.final_type())"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 58,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[0 0 0 0 0 0 0 0]\n",
+       "[0 0 0 0 0 0 0 0]\n",
+       "[0 0 0 0 0 0 0 0]\n",
+       "[0 0 0 0 0 0 0 0]\n",
+       "[0 0 0 4 2 3 3 2]\n",
+       "[4 0 4 4 2 0 0 4]\n",
+       "[0 0 4 3 0 0 1 2]\n",
+       "[3 0 0 0 0 1 0 0]"
+      ]
+     },
+     "execution_count": 58,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "C.frobenius_matrix()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 62,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "test [0, -1, -1, -1, 4, -1, -1, -1, 4]\n",
+      "poprawiony ft [0, 1, 2, 3, 4, 4, 4, 4, 4]\n"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "[1, 2, 3, 4]"
+      ]
+     },
+     "execution_count": 62,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "flag(F, V, p)"
    ]
   },
   {