diff --git a/superelliptic_alpha.ipynb b/superelliptic_alpha.ipynb
index 8cb0a73..e187325 100644
--- a/superelliptic_alpha.ipynb
+++ b/superelliptic_alpha.ipynb
@@ -2,21 +2,16 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
+   "execution_count": 6,
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "Exiting Sage (CPU time 0m6.48s, Wall time 11m10.64s).\n"
-     ]
-    }
    ],
    "source": [
     "def basis_holomorphic_differentials_degree(f, m, p):\n",
     "    kxi1.<xi1> = PolynomialRing(GF(p))\n",
-    "    kxi = kxi.quotient(xi1^p)\n",
+    "    kxi = kxi1.quotient(xi1^p)\n",
     "    xi = kxi(xi1)\n",
     "    r = f.degree()\n",
     "    delta = GCD(r, m)\n",
@@ -50,7 +45,7 @@
     "    r = f.degree()\n",
     "    delta = GCD(r, m)\n",
     "    kxi1.<xi1> = PolynomialRing(GF(p))\n",
-    "    kxi = kxi.quotient(xi1^p)\n",
+    "    kxi = kxi1.quotient(xi1^p)\n",
     "    xi = kxi(xi1)\n",
     "    Rx.<x> = PolynomialRing(kxi)\n",
     "    Rxy.<x, y> = PolynomialRing(kxi, 2)\n",
@@ -92,7 +87,7 @@
     "    \n",
     "    def __init__(self, f, m, p):\n",
     "        kxi1.<xi1> = PolynomialRing(GF(p))\n",
-    "        kxi = kxi.quotient(xi1^p)\n",
+    "        kxi = kxi1.quotient(xi1^p)\n",
     "        xi = kxi(xi1)\n",
     "        Rx.<x> = PolynomialRing(kxi)\n",
     "        Rxy.<x, y> = PolynomialRing(kxi, 2)\n",
@@ -168,7 +163,7 @@
     "        g = self.genus()\n",
     "        p = self.characteristic\n",
     "        kxi1.<xi1> = PolynomialRing(GF(p))\n",
-    "        kxi = kxi.quotient(xi1^p)\n",
+    "        kxi = kxi1.quotient(xi1^p)\n",
     "        xi = kxi(xi1)\n",
     "        M = matrix(kxi, g, g)\n",
     "        for i in range(0, len(basis)):\n",
@@ -189,7 +184,7 @@
     "def reduction(C, g):\n",
     "    p = C.characteristic\n",
     "    kxi1.<xi1> = PolynomialRing(GF(p))\n",
-    "    kxi = kxi.quotient(xi1^p)\n",
+    "    kxi = kxi1.quotient(xi1^p)\n",
     "    xi = kxi(xi1)\n",
     "    Rxy.<x, y> = PolynomialRing(kxi, 2)\n",
     "    Fxy = FractionField(Rxy)\n",
@@ -217,7 +212,7 @@
     "def reduction_form(C, g):\n",
     "    p = C.characteristic\n",
     "    kxi1.<xi1> = PolynomialRing(GF(p))\n",
-    "    kxi = kxi.quotient(xi1^p)\n",
+    "    kxi = kxi1.quotient(xi1^p)\n",
     "    xi = kxi(xi1)\n",
     "    Rxy.<x, y> = PolynomialRing(kxi, 2)\n",
     "    Fxy = FractionField(Rxy)\n",
@@ -245,7 +240,7 @@
     "    def __init__(self, C, g):\n",
     "        p = C.characteristic\n",
     "        kxi1.<xi1> = PolynomialRing(GF(p))\n",
-    "        kxi = kxi.quotient(xi1^p)\n",
+    "        kxi = kxi1.quotient(xi1^p)\n",
     "        xi = kxi(xi1)\n",
     "        Rxy.<x, y> = PolynomialRing(kxi, 2)\n",
     "        Fxy = FractionField(Rxy)\n",
@@ -589,9 +584,12 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
+   "execution_count": 7,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
    "source": [
     "def preimage(U, V, M): #preimage of subspace U under M\n",
     "    basis_preimage = M.right_kernel().basis()\n",
@@ -678,18 +676,44 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
+   "execution_count": 8,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "ename": "SyntaxError",
+     "evalue": "invalid syntax (925769656.py, line 1)",
+     "output_type": "error",
+     "traceback": [
+      "\u001b[0;36m  File \u001b[0;32m\"/tmp/ipykernel_968/925769656.py\"\u001b[0;36m, line \u001b[0;32m1\u001b[0m\n\u001b[0;31m    def auto_a()\u001b[0m\n\u001b[0m                ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m invalid syntax\n"
+     ]
+    }
+   ],
+   "source": [
+    "def auto_a()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 0,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
    "source": [
     "from datetime import datetime"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
+   "execution_count": 0,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
    "source": [
     "now = datetime.now()\n",
     "\n",
@@ -710,8 +734,11 @@
   {
    "cell_type": "code",
    "execution_count": 15,
-   "metadata": {},
-   "outputs": [],
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
    "source": [
     "def p_cov(C):\n",
     "    m = C.exponent\n",
@@ -724,7 +751,7 @@
    "cell_type": "code",
    "execution_count": 26,
    "metadata": {
-    "scrolled": false
+    "collapsed": false
    },
    "outputs": [
     {
@@ -746,7 +773,9 @@
   {
    "cell_type": "code",
    "execution_count": 29,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "name": "stdout",
@@ -769,7 +798,9 @@
   {
    "cell_type": "code",
    "execution_count": 24,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -778,7 +809,8 @@
       ]
      },
      "execution_count": 24,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -789,7 +821,9 @@
   {
    "cell_type": "code",
    "execution_count": 10,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "name": "stdout",
@@ -938,7 +972,9 @@
   {
    "cell_type": "code",
    "execution_count": 7,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "name": "stdout",
@@ -958,6 +994,7 @@
    "cell_type": "code",
    "execution_count": 81,
    "metadata": {
+    "collapsed": false,
     "scrolled": true
    },
    "outputs": [
@@ -982,7 +1019,9 @@
   {
    "cell_type": "code",
    "execution_count": 58,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -998,7 +1037,8 @@
       ]
      },
      "execution_count": 58,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1009,7 +1049,9 @@
   {
    "cell_type": "code",
    "execution_count": 62,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "name": "stdout",
@@ -1026,7 +1068,8 @@
       ]
      },
      "execution_count": 62,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1038,6 +1081,7 @@
    "cell_type": "code",
    "execution_count": 24,
    "metadata": {
+    "collapsed": false,
     "scrolled": true
    },
    "outputs": [
@@ -1063,7 +1107,9 @@
   {
    "cell_type": "code",
    "execution_count": 26,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "name": "stdout",
@@ -1084,7 +1130,9 @@
   {
    "cell_type": "code",
    "execution_count": 31,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1093,7 +1141,8 @@
       ]
      },
      "execution_count": 31,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1104,7 +1153,9 @@
   {
    "cell_type": "code",
    "execution_count": 73,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1115,7 +1166,8 @@
       ]
      },
      "execution_count": 73,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1129,8 +1181,11 @@
   {
    "cell_type": "code",
    "execution_count": 77,
-   "metadata": {},
-   "outputs": [],
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
    "source": [
     "l = U.basis()\n",
     "l = l +[(1, 1/3)]"
@@ -1139,8 +1194,11 @@
   {
    "cell_type": "code",
    "execution_count": 24,
-   "metadata": {},
-   "outputs": [],
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
    "source": [
     "###fragment kodu do obliczania residuuow w niesk - zaniechany\n",
     "#def more_v(f, prec):\n",
@@ -1170,7 +1228,9 @@
   {
    "cell_type": "code",
    "execution_count": 211,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "name": "stdout",
@@ -1194,15 +1254,21 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": []
+   "execution_count": 0,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
+   "source": [
+   ]
   },
   {
    "cell_type": "code",
    "execution_count": 212,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1218,7 +1284,8 @@
       ]
      },
      "execution_count": 212,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1230,7 +1297,7 @@
    "cell_type": "code",
    "execution_count": 213,
    "metadata": {
-    "scrolled": false
+    "collapsed": false
    },
    "outputs": [
     {
@@ -1247,7 +1314,8 @@
       ]
      },
      "execution_count": 213,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1258,8 +1326,11 @@
   {
    "cell_type": "code",
    "execution_count": 214,
-   "metadata": {},
-   "outputs": [],
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
    "source": [
     "A = C.frobenius_matrix()\n",
     "B = C.verschiebung_matrix()"
@@ -1268,7 +1339,9 @@
   {
    "cell_type": "code",
    "execution_count": 227,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1277,7 +1350,8 @@
       ]
      },
      "execution_count": 227,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1288,7 +1362,9 @@
   {
    "cell_type": "code",
    "execution_count": 228,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1297,7 +1373,8 @@
       ]
      },
      "execution_count": 228,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1308,7 +1385,9 @@
   {
    "cell_type": "code",
    "execution_count": 225,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1317,7 +1396,8 @@
       ]
      },
      "execution_count": 225,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1328,8 +1408,11 @@
   {
    "cell_type": "code",
    "execution_count": 83,
-   "metadata": {},
-   "outputs": [],
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
    "source": [
     "omega = diffn(superelliptic_function(C, y^2))"
    ]
@@ -1337,7 +1420,9 @@
   {
    "cell_type": "code",
    "execution_count": 84,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1346,7 +1431,8 @@
       ]
      },
      "execution_count": 84,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1357,7 +1443,9 @@
   {
    "cell_type": "code",
    "execution_count": 85,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1366,7 +1454,8 @@
       ]
      },
      "execution_count": 85,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1384,8 +1473,11 @@
   {
    "cell_type": "code",
    "execution_count": 86,
-   "metadata": {},
-   "outputs": [],
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
    "source": [
     "H = HyperellipticCurve(x^5 - x + 1)"
    ]
@@ -1393,7 +1485,9 @@
   {
    "cell_type": "code",
    "execution_count": 40,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1402,7 +1496,8 @@
       ]
      },
      "execution_count": 40,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1413,8 +1508,11 @@
   {
    "cell_type": "code",
    "execution_count": 84,
-   "metadata": {},
-   "outputs": [],
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
    "source": [
     "f = x^3 + x + 2"
    ]
@@ -1422,7 +1520,9 @@
   {
    "cell_type": "code",
    "execution_count": 86,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1431,7 +1531,8 @@
       ]
      },
      "execution_count": 86,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1442,8 +1543,11 @@
   {
    "cell_type": "code",
    "execution_count": 3,
-   "metadata": {},
-   "outputs": [],
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
    "source": [
     "p = 5\n",
     "R1.<x> = PolynomialRing(GF(p))\n",
@@ -1456,7 +1560,9 @@
   {
    "cell_type": "code",
    "execution_count": 4,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1465,7 +1571,8 @@
       ]
      },
      "execution_count": 4,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1477,7 +1584,9 @@
   {
    "cell_type": "code",
    "execution_count": 57,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1486,7 +1595,8 @@
       ]
      },
      "execution_count": 57,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1497,7 +1607,9 @@
   {
    "cell_type": "code",
    "execution_count": 62,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "ename": "AttributeError",
@@ -1521,7 +1633,9 @@
   {
    "cell_type": "code",
    "execution_count": 35,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1530,7 +1644,8 @@
       ]
      },
      "execution_count": 35,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1541,7 +1656,9 @@
   {
    "cell_type": "code",
    "execution_count": 36,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1550,7 +1667,8 @@
       ]
      },
      "execution_count": 36,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1561,8 +1679,11 @@
   {
    "cell_type": "code",
    "execution_count": 37,
-   "metadata": {},
-   "outputs": [],
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
    "source": [
     "R.<x> = PolynomialRing(GF(5))"
    ]
@@ -1570,8 +1691,11 @@
   {
    "cell_type": "code",
    "execution_count": 39,
-   "metadata": {},
-   "outputs": [],
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
    "source": [
     "R = (x^3+x).parent()"
    ]
@@ -1579,8 +1703,11 @@
   {
    "cell_type": "code",
    "execution_count": 44,
-   "metadata": {},
-   "outputs": [],
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
    "source": [
     "R.<x, y> = PolynomialRing(GF(5))\n",
     "RR = FractionField(R)\n",
@@ -1590,7 +1717,9 @@
   {
    "cell_type": "code",
    "execution_count": 45,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1599,7 +1728,8 @@
       ]
      },
      "execution_count": 45,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1610,8 +1740,11 @@
   {
    "cell_type": "code",
    "execution_count": 42,
-   "metadata": {},
-   "outputs": [],
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
    "source": [
     "dict1 = {}\n",
     "dict1[3] = 5\n",
@@ -1621,8 +1754,11 @@
   {
    "cell_type": "code",
    "execution_count": 46,
-   "metadata": {},
-   "outputs": [],
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
    "source": [
     "degrees1_inv = {b:a for a, b in dict1.items()}"
    ]
@@ -1630,7 +1766,9 @@
   {
    "cell_type": "code",
    "execution_count": 47,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1639,7 +1777,8 @@
       ]
      },
      "execution_count": 47,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1650,7 +1789,9 @@
   {
    "cell_type": "code",
    "execution_count": 28,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1659,7 +1800,8 @@
       ]
      },
      "execution_count": 28,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1670,8 +1812,11 @@
   {
    "cell_type": "code",
    "execution_count": 29,
-   "metadata": {},
-   "outputs": [],
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
    "source": [
     "basis = C.basis_de_rham()"
    ]
@@ -1679,7 +1824,9 @@
   {
    "cell_type": "code",
    "execution_count": 32,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1688,7 +1835,8 @@
       ]
      },
      "execution_count": 32,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1699,7 +1847,9 @@
   {
    "cell_type": "code",
    "execution_count": 9,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1708,7 +1858,8 @@
       ]
      },
      "execution_count": 9,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1723,7 +1874,9 @@
   {
    "cell_type": "code",
    "execution_count": 10,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1732,7 +1885,8 @@
       ]
      },
      "execution_count": 10,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1743,8 +1897,11 @@
   {
    "cell_type": "code",
    "execution_count": 11,
-   "metadata": {},
-   "outputs": [],
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
    "source": [
     "Fpbar = GF(5).algebraic_closure()\n",
     "z = Fpbar.zeta(7)"
@@ -1753,7 +1910,9 @@
   {
    "cell_type": "code",
    "execution_count": 23,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "ename": "TypeError",
@@ -1780,7 +1939,9 @@
   {
    "cell_type": "code",
    "execution_count": 21,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1789,7 +1950,8 @@
       ]
      },
      "execution_count": 21,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1800,8 +1962,11 @@
   {
    "cell_type": "code",
    "execution_count": 7,
-   "metadata": {},
-   "outputs": [],
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
    "source": [
     "Rx.<x> = PolynomialRing(QQ)\n",
     "f = sum((i+1)*x^i for i in range(0, 10))"
@@ -1810,7 +1975,9 @@
   {
    "cell_type": "code",
    "execution_count": 8,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1819,7 +1986,8 @@
       ]
      },
      "execution_count": 8,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1830,7 +1998,9 @@
   {
    "cell_type": "code",
    "execution_count": 9,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1839,7 +2009,8 @@
       ]
      },
      "execution_count": 9,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1850,7 +2021,9 @@
   {
    "cell_type": "code",
    "execution_count": 45,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "name": "stdout",
@@ -1870,7 +2043,9 @@
   {
    "cell_type": "code",
    "execution_count": 41,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1879,7 +2054,8 @@
       ]
      },
      "execution_count": 41,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1890,7 +2066,9 @@
   {
    "cell_type": "code",
    "execution_count": 42,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1899,16 +2077,20 @@
       ]
      },
      "execution_count": 42,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
-   "source": []
+   "source": [
+   ]
   },
   {
    "cell_type": "code",
    "execution_count": 30,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "name": "stdout",
@@ -1940,8 +2122,11 @@
   {
    "cell_type": "code",
    "execution_count": 1,
-   "metadata": {},
-   "outputs": [],
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
    "source": [
     "kxi1.<xi1> = PolynomialRing(GF(p))\n",
     "kxi = kxi.quotient(xi1^p)\n",
@@ -1951,7 +2136,9 @@
   {
    "cell_type": "code",
    "execution_count": 2,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1960,7 +2147,8 @@
       ]
      },
      "execution_count": 2,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -1971,8 +2159,11 @@
   {
    "cell_type": "code",
    "execution_count": 3,
-   "metadata": {},
-   "outputs": [],
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
    "source": [
     "s = kxi2(xi)"
    ]
@@ -1980,7 +2171,9 @@
   {
    "cell_type": "code",
    "execution_count": 4,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -1989,7 +2182,8 @@
       ]
      },
      "execution_count": 4,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -2000,7 +2194,9 @@
   {
    "cell_type": "code",
    "execution_count": 5,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -2009,7 +2205,8 @@
       ]
      },
      "execution_count": 5,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -2020,8 +2217,11 @@
   {
    "cell_type": "code",
    "execution_count": 8,
-   "metadata": {},
-   "outputs": [],
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
    "source": [
     "a = 1+2*s+s^2"
    ]
@@ -2029,7 +2229,9 @@
   {
    "cell_type": "code",
    "execution_count": 10,
-   "metadata": {},
+   "metadata": {
+    "collapsed": false
+   },
    "outputs": [
     {
      "data": {
@@ -2038,7 +2240,8 @@
       ]
      },
      "execution_count": 10,
-     "metadata": {},
+     "metadata": {
+     },
      "output_type": "execute_result"
     }
    ],
@@ -2048,17 +2251,65 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": []
+   "execution_count": 10,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "ename": "NotImplementedError",
+     "evalue": "",
+     "output_type": "error",
+     "traceback": [
+      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
+      "\u001b[0;31mNotImplementedError\u001b[0m                       Traceback (most recent call last)",
+      "\u001b[0;32m/tmp/ipykernel_968/1691347420.py\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m      8\u001b[0m \u001b[0mf\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mf1\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\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      9\u001b[0m \u001b[0mm\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[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 10\u001b[0;31m \u001b[0mC\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msuperelliptic\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mm\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[0m",
+      "\u001b[0;32m/tmp/ipykernel_968/2781228428.py\u001b[0m in \u001b[0;36m__init__\u001b[0;34m(self, f, m, p)\u001b[0m\n\u001b[1;32m     81\u001b[0m         \u001b[0mRx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mPolynomialRing\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mkxi\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[1;32m     82\u001b[0m         \u001b[0mRxy\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mPolynomialRing\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mkxi\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[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'y'\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[0my\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mRxy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_first_ngens\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 83\u001b[0;31m         \u001b[0mFxy\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mFractionField\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mRxy\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     84\u001b[0m         \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpolynomial\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mRx\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[1;32m     85\u001b[0m         \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexponent\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mm\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/rings/fraction_field.py\u001b[0m in \u001b[0;36mFractionField\u001b[0;34m(R, names)\u001b[0m\n\u001b[1;32m    129\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mring\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Ring\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mR\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    130\u001b[0m         \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"R must be a ring\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 131\u001b[0;31m     \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mR\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_integral_domain\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    132\u001b[0m         \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"R must be an integral domain.\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    133\u001b[0m     \u001b[0;32mreturn\u001b[0m \u001b[0mR\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfraction_field\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/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/rings/polynomial/multi_polynomial_ring_base.pyx\u001b[0m in \u001b[0;36msage.rings.polynomial.multi_polynomial_ring_base.MPolynomialRing_base.is_integral_domain (build/cythonized/sage/rings/polynomial/multi_polynomial_ring_base.c:4294)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    107\u001b[0m             \u001b[0;32mFalse\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    108\u001b[0m         \"\"\"\n\u001b[0;32m--> 109\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbase_ring\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_integral_domain\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mproof\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    110\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    111\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0mis_noetherian\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/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/rings/ring.pyx\u001b[0m in \u001b[0;36msage.rings.ring.Ring.is_integral_domain (build/cythonized/sage/rings/ring.c:8605)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    999\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1000\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mproof\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1001\u001b[0;31m             \u001b[0;32mraise\u001b[0m \u001b[0mNotImplementedError\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1002\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   1003\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0;32mFalse\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;31mNotImplementedError\u001b[0m: "
+     ]
+    }
+   ],
+   "source": [
+    "p = 5\n",
+    "kxi1.<xi1> = PolynomialRing(GF(p))\n",
+    "kxi = kxi1.quotient(xi1^p)\n",
+    "xi = kxi(xi1)\n",
+    "Rx.<x> = PolynomialRing(kxi)\n",
+    "\n",
+    "f1 = x^3 + x + 1\n",
+    "f = f1(x^p)\n",
+    "m = 2\n",
+    "C = superelliptic(f, m, p)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 0,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
+   "source": [
+   ]
   }
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "SageMath 9.1",
-   "language": "sage",
-   "name": "sagemath"
+   "display_name": "SageMath 9.5",
+   "language": "sagemath",
+   "metadata": {
+    "cocalc": {
+     "description": "Open-source mathematical software system",
+     "priority": 10,
+     "url": "https://www.sagemath.org/"
+    }
+   },
+   "name": "sage-9.5",
+   "resource_dir": "/ext/jupyter/kernels/sage-9.5"
   },
   "language_info": {
    "codemirror_mode": {
@@ -2070,9 +2321,9 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.7.3"
+   "version": "3.9.9"
   }
  },
  "nbformat": 4,
- "nbformat_minor": 2
-}
+ "nbformat_minor": 4
+}
\ No newline at end of file