5-1-1-Elliptic-Curve.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "1d7dce20",
   "metadata": {},
   "outputs": [],
   "source": [
    "F5 = GF(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "793dffe5",
   "metadata": {},
   "outputs": [],
   "source": [
    "a = F5(2)\n",
    "b = F5(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "e9175800",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# check discriminant\n",
    "F5(6) * (F5(4) * a^3 + F5(27) * b^2) != F5(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "a142f24a",
   "metadata": {},
   "outputs": [],
   "source": [
    "E = EllipticCurve(F5, [a,b]) # y^2 == x^3 + ax + b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "e96e095c",
   "metadata": {},
   "outputs": [],
   "source": [
    "P = E(0,2) # 2^2 == 0^3 + 2*0 + 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "f0a9aa66",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 2)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.xy()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "ffb8dba8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2, 4)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P = E(2,4)\n",
    "P.xy()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "947455b9",
   "metadata": {},
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "Coordinates [3, 1, 1] do not define a point on Elliptic Curve defined by y^2 = x^3 + 2*x + 4 over Finite Field of size 5",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mKeyError\u001b[0m                                  Traceback (most recent call last)",
      "File \u001b[0;32m/private/var/tmp/sage-9.8-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/structure/category_object.pyx:839\u001b[0m, in \u001b[0;36msage.structure.category_object.CategoryObject.getattr_from_category (build/cythonized/sage/structure/category_object.c:7204)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    838\u001b[0m try:\n\u001b[0;32m--> 839\u001b[0m     return self.__cached_methods[name]\n\u001b[1;32m    840\u001b[0m except KeyError:\n",
      "\u001b[0;31mKeyError\u001b[0m: 'point_homset'",
      "\nDuring handling of the above exception, another exception occurred:\n",
      "\u001b[0;31mAttributeError\u001b[0m                            Traceback (most recent call last)",
      "File \u001b[0;32m/private/var/tmp/sage-9.8-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/schemes/projective/projective_subscheme.py:122\u001b[0m, in \u001b[0;36mAlgebraicScheme_subscheme_projective.point\u001b[0;34m(self, v, check)\u001b[0m\n\u001b[1;32m    121\u001b[0m \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[0;32m--> 122\u001b[0m     \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_point\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mpoint_homset\u001b[49m\u001b[43m(\u001b[49m\u001b[43m)\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mv\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mcheck\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mcheck\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    123\u001b[0m \u001b[38;5;28;01mexcept\u001b[39;00m \u001b[38;5;167;01mAttributeError\u001b[39;00m:  \u001b[38;5;66;03m# legacy code without point_homset\u001b[39;00m\n",
      "File \u001b[0;32m/private/var/tmp/sage-9.8-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/schemes/elliptic_curves/ell_point.py:256\u001b[0m, in \u001b[0;36mEllipticCurvePoint_field.__init__\u001b[0;34m(self, curve, v, check)\u001b[0m\n\u001b[1;32m    235\u001b[0m \u001b[38;5;124;03m\"\"\"\u001b[39;00m\n\u001b[1;32m    236\u001b[0m \u001b[38;5;124;03mConstructor for a point on an elliptic curve.\u001b[39;00m\n\u001b[1;32m    237\u001b[0m \n\u001b[0;32m   (...)\u001b[0m\n\u001b[1;32m    254\u001b[0m \u001b[38;5;124;03m    (1 : -2 : 1)\u001b[39;00m\n\u001b[1;32m    255\u001b[0m \u001b[38;5;124;03m\"\"\"\u001b[39;00m\n\u001b[0;32m--> 256\u001b[0m point_homset \u001b[38;5;241m=\u001b[39m \u001b[43mcurve\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mpoint_homset\u001b[49m()\n\u001b[1;32m    257\u001b[0m R \u001b[38;5;241m=\u001b[39m point_homset\u001b[38;5;241m.\u001b[39mvalue_ring()\n",
      "File \u001b[0;32m/private/var/tmp/sage-9.8-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/structure/category_object.pyx:833\u001b[0m, in \u001b[0;36msage.structure.category_object.CategoryObject.__getattr__ (build/cythonized/sage/structure/category_object.c:7123)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    832\u001b[0m \"\"\"\n\u001b[0;32m--> 833\u001b[0m return self.getattr_from_category(name)\n\u001b[1;32m    834\u001b[0m \n",
      "File \u001b[0;32m/private/var/tmp/sage-9.8-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/structure/category_object.pyx:848\u001b[0m, in \u001b[0;36msage.structure.category_object.CategoryObject.getattr_from_category (build/cythonized/sage/structure/category_object.c:7289)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    847\u001b[0m \n\u001b[0;32m--> 848\u001b[0m             attr = getattr_from_other_class(self, cls, name)\n\u001b[1;32m    849\u001b[0m             self.__cached_methods[name] = attr\n",
      "File \u001b[0;32m/private/var/tmp/sage-9.8-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/cpython/getattr.pyx:356\u001b[0m, in \u001b[0;36msage.cpython.getattr.getattr_from_other_class (build/cythonized/sage/cpython/getattr.c:2706)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    355\u001b[0m     dummy_error_message.name = name\n\u001b[0;32m--> 356\u001b[0m     raise AttributeError(dummy_error_message)\n\u001b[1;32m    357\u001b[0m cdef PyObject* attr = instance_getattr(cls, name)\n",
      "\u001b[0;31mAttributeError\u001b[0m: 'sage.rings.finite_rings.integer_mod.IntegerMod_int' object has no attribute '__custom_name'",
      "\nDuring handling of the above exception, another exception occurred:\n",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "Cell \u001b[0;32mIn [13], line 1\u001b[0m\n\u001b[0;32m----> 1\u001b[0m P \u001b[38;5;241m=\u001b[39m \u001b[43mE\u001b[49m\u001b[43m(\u001b[49m\u001b[43mInteger\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m3\u001b[39;49m\u001b[43m)\u001b[49m\u001b[43m,\u001b[49m\u001b[43mInteger\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m1\u001b[39;49m\u001b[43m)\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m      2\u001b[0m P\u001b[38;5;241m.\u001b[39mxy()\n",
      "File \u001b[0;32m/private/var/tmp/sage-9.8-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/schemes/elliptic_curves/ell_generic.py:570\u001b[0m, in \u001b[0;36mEllipticCurve_generic.__call__\u001b[0;34m(self, *args, **kwds)\u001b[0m\n\u001b[1;32m    567\u001b[0m             \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_reduce_point(args[\u001b[38;5;241m0\u001b[39m], characteristic)\n\u001b[1;32m    568\u001b[0m     args \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mtuple\u001b[39m(args[\u001b[38;5;241m0\u001b[39m])\n\u001b[0;32m--> 570\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43mplane_curve\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mProjectivePlaneCurve\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[38;5;21;43m__call__\u001b[39;49m\u001b[43m(\u001b[49m\u001b[38;5;28;43mself\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwds\u001b[49m\u001b[43m)\u001b[49m\n",
      "File \u001b[0;32m/private/var/tmp/sage-9.8-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/schemes/generic/scheme.py:266\u001b[0m, in \u001b[0;36mScheme.__call__\u001b[0;34m(self, *args)\u001b[0m\n\u001b[1;32m    264\u001b[0m             \u001b[38;5;28;01mreturn\u001b[39;00m S\n\u001b[1;32m    265\u001b[0m         args \u001b[38;5;241m=\u001b[39m S\n\u001b[0;32m--> 266\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mpoint\u001b[49m\u001b[43m(\u001b[49m\u001b[43margs\u001b[49m\u001b[43m)\u001b[49m\n",
      "File \u001b[0;32m/private/var/tmp/sage-9.8-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/schemes/projective/projective_subscheme.py:124\u001b[0m, in \u001b[0;36mAlgebraicScheme_subscheme_projective.point\u001b[0;34m(self, v, check)\u001b[0m\n\u001b[1;32m    122\u001b[0m         \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_point(\u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mpoint_homset(), v, check\u001b[38;5;241m=\u001b[39mcheck)\n\u001b[1;32m    123\u001b[0m     \u001b[38;5;28;01mexcept\u001b[39;00m \u001b[38;5;167;01mAttributeError\u001b[39;00m:  \u001b[38;5;66;03m# legacy code without point_homset\u001b[39;00m\n\u001b[0;32m--> 124\u001b[0m         \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_point\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;28;43mself\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mv\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mcheck\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mcheck\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    126\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mpoint_homset()(v, check\u001b[38;5;241m=\u001b[39mcheck)\n",
      "File \u001b[0;32m/private/var/tmp/sage-9.8-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/schemes/elliptic_curves/ell_point.py:295\u001b[0m, in \u001b[0;36mEllipticCurvePoint_field.__init__\u001b[0;34m(self, curve, v, check)\u001b[0m\n\u001b[1;32m    293\u001b[0m         test \u001b[38;5;241m=\u001b[39m y\u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39m\u001b[38;5;241m2\u001b[39m \u001b[38;5;241m+\u001b[39m (a1\u001b[38;5;241m*\u001b[39mx\u001b[38;5;241m+\u001b[39ma3)\u001b[38;5;241m*\u001b[39my \u001b[38;5;241m-\u001b[39m (((x\u001b[38;5;241m+\u001b[39ma2)\u001b[38;5;241m*\u001b[39mx\u001b[38;5;241m+\u001b[39ma4)\u001b[38;5;241m*\u001b[39mx\u001b[38;5;241m+\u001b[39ma6)\n\u001b[1;32m    294\u001b[0m     \u001b[38;5;28;01mif\u001b[39;00m \u001b[38;5;129;01mnot\u001b[39;00m test \u001b[38;5;241m==\u001b[39m \u001b[38;5;241m0\u001b[39m:\n\u001b[0;32m--> 295\u001b[0m         \u001b[38;5;28;01mraise\u001b[39;00m \u001b[38;5;167;01mTypeError\u001b[39;00m(\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mCoordinates \u001b[39m\u001b[38;5;132;01m%s\u001b[39;00m\u001b[38;5;124m do not define a point on \u001b[39m\u001b[38;5;132;01m%s\u001b[39;00m\u001b[38;5;124m\"\u001b[39m \u001b[38;5;241m%\u001b[39m (\u001b[38;5;28mlist\u001b[39m(v), curve))\n\u001b[1;32m    297\u001b[0m SchemeMorphism_point_abelian_variety_field\u001b[38;5;241m.\u001b[39m\u001b[38;5;21m__init__\u001b[39m(\u001b[38;5;28mself\u001b[39m, point_homset, v, check\u001b[38;5;241m=\u001b[39m\u001b[38;5;28;01mFalse\u001b[39;00m)\n",
      "\u001b[0;31mTypeError\u001b[0m: Coordinates [3, 1, 1] do not define a point on Elliptic Curve defined by y^2 = x^3 + 2*x + 4 over Finite Field of size 5"
     ]
    }
   ],
   "source": [
    "# P = E(3,1) # this is not in the curve, so raises an error."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "af01b50e",
   "metadata": {},
   "outputs": [],
   "source": [
    "INF = E(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "3d4779ca",
   "metadata": {},
   "outputs": [],
   "source": [
    "try:\n",
    "    INF.xy()\n",
    "except ZeroDivisionError:\n",
    "    pass"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "a91db892",
   "metadata": {},
   "outputs": [],
   "source": [
    "P = E.plot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "adf949c8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Tpjh27FjMmzcvjhw5Mmzf3bt3x4EDB/q3a6+9dsRFAwAQ0VJL41deeWXA/vPPPx8TJkyI119/PW655ZbT9p0wYUJcdtllNRcIAMDgzuo7cpVKJSIirrjiimHb3njjjdHW1hZz586NzZs3n7ZtX19fVKvVARsAAAONOMhlWRaLFy+Om2++Oa6//voh27W1tcXKlSujq6sr1q5dG9OnT4+5c+fG1q1bh+zT2dkZhUKhf5syZcpIywQAuGDlsizLRtJx0aJFsWHDhvjpT38akydPrqnvggULIpfLxfr16wd9va+vL/r6+vr3q9VqTJkyJSqVSuTz+ZGUCwBwwRnRitxDDz0U69evj82bN9cc4iIibrrppnj33XeHfL21tTXy+fyADQCAgWq62SHLsnjooYdi3bp1sWXLlpg6deqITtrT0xNtbW0j6gsAwAk1BblFixbF97///fjhD38Y48aNi97e3oiIKBQKcckll0RExNKlS2P//v3xwgsvRETE008/HVdffXXMmDEjjh49GqtXr46urq7o6uoa5Y8CANBcagpyy5cvj4iIz3zmMwOOP//883H//fdHRMSBAwdi7969/a8dPXo0lixZEvv3749LLrkkZsyYERs2bIj58+efXeUAAE1uxDc7nE/VajUKhYKbHQAAfo/fWgUASJQgBwCQKEEOACBRghwAQKIEOQCARAlyAACJauggVy6Xo1gsRqlUqncpAAANx3PkAAAS1dArcgAADE2QAwBIlCAHAJAoQQ4AIFGCHABAogQ5AIBECXIAAIkS5AAAEiXIAQAkSpADAEiUIAcAkChBDgAgUQ0d5MrlchSLxSiVSvUuBQCg4eSyLMvqXcRwqtVqFAqFqFQqkc/n610OAEBDaOgVOQAAhibIAQAkSpADAEiUIAcAkChBDgAgUYIcAECiBDkAgEQJcgAAiRLkAAASJcgBACRKkAMASFRDB7lyuRzFYjFKpVK9SwEAaDi5LMuyehcxnGq1GoVCISqVSuTz+XqXAwDQEBp6RQ4AgKEJcgAAiRLkAAASJcgBACRKkAMASJQgBwCQKEEOACBRghwAQKIEOQCARAlyAACJEuQAABLV0EGuXC5HsViMUqlU71IAABpOLsuyrN5FDKdarUahUIhKpRL5fL7e5QAANISGXpEDAGBoghwAQKIEOQCARAlyAACJEuQAABIlyAEAJEqQAwBIVE1BrrOzM0qlUowbNy4mTJgQd911V+zevXvYft3d3TFr1qwYO3ZsTJs2LVasWDHiggEAOKGmINfd3R2LFi2KV199NTZt2hTHjh2LefPmxZEjR4bss2fPnpg/f37MmTMnenp6YtmyZfHwww9HV1fXWRcPANDMzuqXHX7961/HhAkToru7O2655ZZB2zz++OOxfv362LVrV/+xjo6OeOONN2Lbtm3DnmPFioinnqrGe+8Voru7Erfc4pcdAAAizvI7cpVKJSIirrjiiiHbbNu2LebNmzfg2O233x7bt2+PDz/8cNA+fX19Ua1W4yc/qcYDD1TjvfeqERFxzz1nUy0AwIVlxEEuy7JYvHhx3HzzzXH99dcP2a63tzcmTpw44NjEiRPj2LFjcejQoUH7dHZ2RqFQiFtuKUREISKmRETE/v0Rx4+PtGIAgAvLiIPcgw8+GG+++Wb84Ac/GLZtLpcbsH/yau5Hj5+0dOnSqFQqsWdPJaZMqUTEvoiI+I//MeKii0ZaMQDAhWVEQe6hhx6K9evXx+bNm2Py5MmnbXvVVVdFb2/vgGMHDx6MlpaWGD9+/KB9WltbI5/Px9VX5+O11/Lx1FMnvhfnZlcAgN+pKchlWRYPPvhgrF27Nn784x/H1KlTh+3T3t4emzZtGnBs48aNMXv27BgzZsyw/SdOjOjoOPFvq3EAAL9TU5BbtGhRrF69Or7//e/HuHHjore3N3p7e+Pf/u3f+tssXbo07rvvvv79jo6OeP/992Px4sWxa9eueO6552LVqlWxZMmS0fsUAABNqKbHjwz1nbbnn38+7r///oiIuP/+++O9996LLVu29L/e3d0djz32WLz11lsxadKkePzxx6Pj5DLbGahWq1EoFKJSqUQ+7/EjAAARZ/kcufNFkAMAOJXfWgUASJQgBwCQKEEOACBRghwAQKIEOQCARAlyAACJauggVy6Xo1gsRqlUqncpAAANx3PkAAAS1dArcgAADE2QAwBIlCAHAJAoQQ4AIFGCHABAogQ5AIBECXIAAIkS5AAAEiXIAQAkSpADAEiUIAcAkChBDgAgUQ0d5MrlchSLxSiVSvUuBQCg4eSyLMvqXcRwqtVqFAqFqFQqkc/n610OAEBDaOgVOQAAhibIAQAkSpADAEiUIAcAkChBDgAgUYIcAECiBDkAgEQJcgAAiRLkAAASJcgBACRKkAMASFRDB7lyuRzFYjFKpVK9SwEAaDi5LMuyehcxnGq1GoVCISqVSuTz+XqXAwDQEBp6RQ4AgKEJcgAAiRLkAAASJcgBACRKkAMASJQgBwCQKEEOACBRghwAQKIEOQCARAlyAACJEuQAABLV0EGuXC5HsViMUqlU71IAABpOLsuyrN5FDKdarUahUIhKpRL5fL7e5QAANISGXpEDAGBoghwAQKIEOQCARAlyAACJEuQAABIlyAEAJEqQAwBIVM1BbuvWrbFgwYKYNGlS5HK5ePHFF0/bfsuWLZHL5U7Z3nnnnZHWDABARLTU2uHIkSNxww03xBe/+MW4++67z7jf7t27BzzM98orr6z11AAA/J6ag9wdd9wRd9xxR80nmjBhQlx22WU19wMAYHDn7TtyN954Y7S1tcXcuXNj8+bNp23b19cX1Wp1wAYAwEDnPMi1tbXFypUro6urK9auXRvTp0+PuXPnxtatW4fs09nZGYVCoX+bMmXKuS4TACA5uSzLshF3zuVi3bp1cdddd9XUb8GCBZHL5WL9+vWDvt7X1xd9fX39+9VqNaZMmRKVSmXA9+wAAJpZXR4/ctNNN8W777475Outra2Rz+cHbAAADFSXINfT0xNtbW31ODUAwAWj5rtWP/jgg/jlL3/Zv79nz57YsWNHXHHFFfHJT34yli5dGvv3748XXnghIiKefvrpuPrqq2PGjBlx9OjRWL16dXR1dUVXV9fofQoAgCZUc5Dbvn17fPazn+3fX7x4cURELFy4ML73ve/FgQMHYu/evf2vHz16NJYsWRL79++PSy65JGbMmBEbNmyI+fPnj0L5AADN66xudjhfqtVqFAoFNzsAAPwev7UKAJAoQQ4AIFGCHABAogQ5AIBECXIAAIkS5AAAEiXIAQAkqqGDXLlcjmKxGKVSqd6lAAA0HA8EBgBIVEOvyAEAMDRBDgAgUYIcAECiBDkAgEQJcgAAiRLkAAASJcgBACRKkAMASJQgBwCQKEEOACBRghwAQKIaOsiVy+UoFotRKpXqXQoAQMPJZVmW1buI4VSr1SgUClGpVCKfz9e7HACAhtDQK3IAAAxNkAMASJQgBwCQKEEOACBRghwAQKIEOQCARAlyAACJEuQAABIlyAEAJEqQAwBIlCAHAJCohg5y5XI5isVilEqlepcCANBwclmWZfUuYjjVajUKhUJUKpXI5/P1LgcAoCE09IocAABDE+QAABIlyAEAJEqQAwBIlCAHAJAoQQ4AIFGCHABAogQ5AIBECXIAAIkS5AAAEiXIAQAkqqGDXLlcjmKxGKVSqd6lAAA0nFyWZVm9ixhOtVqNQqEQlUol8vl8vcsBAGgIDb0iBwDA0AQ5AIBECXIAAIkS5AAAEiXIAQAkSpADAEiUIAcAkKiag9zWrVtjwYIFMWnSpMjlcvHiiy8O26e7uztmzZoVY8eOjWnTpsWKFStGUisAAL+n5iB35MiRuOGGG+LZZ589o/Z79uyJ+fPnx5w5c6KnpyeWLVsWDz/8cHR1ddVcLAAAv3NWv+yQy+Vi3bp1cddddw3Z5vHHH4/169fHrl27+o91dHTEG2+8Edu2bTuj8/hlB+Bcev75iLVrI/7ojyK+9rWIj3+83hUBF5LjxyP+23+L+J//M+LWWyOWLInI5UbnvVtG522Gtm3btpg3b96AY7fffnusWrUqPvzwwxgzZswpffr6+qKvr69/v1qtnusygSb13/97xJe+9Lv9SiXiu9+tXz3AhaezM+I//+cT/96wIeKSSyIefHB03vuc3+zQ29sbEydOHHBs4sSJcezYsTh06NCgfTo7O6NQKPRvU6ZMOddlAk3qtdcG7v/TP9WnDuDC9dG/Mx/dPxvn5a7V3EfWD09ezf3o8ZOWLl0alUqlf9u3b985rxFoTrfeOvASx2c/W79agAvTZz5z+v2zcc4vrV511VXR29s74NjBgwejpaUlxo8fP2if1tbWaG1tPdelAcTnPhfxwx9GrFt34jtyS5bUuyLgQvPYYycup/7TP0XcckvE/feP3nuf8yDX3t4eP/rRjwYc27hxY8yePXvQ78cBnG8LFpzYAM6Vjo4T22ir+dLqBx98EDt27IgdO3ZExInHi+zYsSP27t0bEScui95333397Ts6OuL999+PxYsXx65du+K5556LVatWxRL/2wsAcFZqXpHbvn17fPb3vkSyePHiiIhYuHBhfO9734sDBw70h7qIiKlTp8ZLL70Ujz32WJTL5Zg0aVI888wzcffdd49C+QAAzeusniN3vniOHADAqfzWKgBAogQ5AIBEJXFpNcuyOHz4cIwbN27IZ88BADSbJIIcAACncmkVACBRghwAQKIEOQCARAlyAACJEuQAABIlyAEAJEqQAwBIlCAHAJAoQQ4AIFGCHABAolrqdeKTv58KAMDghvud+boFucOHD0ehUKjX6QEAGl6lUol8Pj/k67ksy7LzWE+/WlbkqtVqTJkyJfbt23faDzOYUqkUr732Ws31pdAvpXE53+dMaWyMy+j2G2nfZhgb4zK6/YzN4IzL4EY6Lg27IpfL5Wr+D5zP52vuc9FFF9XcJ6V+EWmMS73OmcLYGJfR7Xe2fS/ksTEuo9vvJGMzOOMyuJGMy+lc8Dc7LFq06ILuN1L1qNPYNEa/kUrp8xmb0e03Uql8vvM9LmdzzlT6jVQqn68ec2Yodbu0WotqtRqFQmHY68TNxrgMzdgMzrgMzdgMzrgMzdgMzrgM7lyNSxIrcq2trfFXf/VX0draWu9SGopxGZqxGZxxGZqxGZxxGZqxGZxxGdy5GpckVuQAADhVEityAACcSpADAEiUIAcAkChBDgAgUQ0T5P7mb/4mpk6dGmPHjo1Zs2bFT37yk9O27+7ujlmzZsXYsWNj2rRpsWLFivNU6flVy7hs2bIlcrncKds777xzHis+97Zu3RoLFiyISZMmRS6XixdffHHYPs0yX2odm2aZM52dnVEqlWLcuHExYcKEuOuuu2L37t3D9rvQ581IxqVZ5szy5ctj5syZ/Q9vbW9vj5dffvm0fS70+RJR+7g0y3z5qM7OzsjlcvHoo4+ett1ozJmGCHL/8A//EI8++mh89atfjZ6enpgzZ07ccccdsXfv3kHb79mzJ+bPnx9z5syJnp6eWLZsWTz88MPR1dV1nis/t2odl5N2794dBw4c6N+uvfba81Tx+XHkyJG44YYb4tlnnz2j9s0yXyJqH5uTLvQ5093dHYsWLYpXX301Nm3aFMeOHYt58+bFkSNHhuzTDPNmJONy0oU+ZyZPnhxPPvlkbN++PbZv3x6f+9zn4s4774y33npr0PbNMF8iah+Xky70+fL7XnvttVi5cmXMnDnztO1Gbc5kDeBP//RPs46OjgHHrrvuuuyJJ54YtP1XvvKV7Lrrrhtw7Mtf/nJ20003nbMa66HWcdm8eXMWEdm//uu/nofqGkNEZOvWrTttm2aZLx91JmPTjHMmy7Ls4MGDWURk3d3dQ7ZpxnlzJuPSrHMmy7Ls8ssvz7773e8O+lozzpeTTjcuzTZfDh8+nF177bXZpk2bsltvvTV75JFHhmw7WnOm7ityR48ejddffz3mzZs34Pi8efPiZz/72aB9tm3bdkr722+/PbZv3x4ffvjhOav1fBrJuJx04403RltbW8ydOzc2b958LstMQjPMl7PVbHOmUqlERMQVV1wxZJtmnDdnMi4nNdOcOX78eKxZsyaOHDkS7e3tg7ZpxvlyJuNyUrPMl0WLFsUXvvCFuO2224ZtO1pzpu5B7tChQ3H8+PGYOHHigOMTJ06M3t7eQfv09vYO2v7YsWNx6NChc1br+TSScWlra4uVK1dGV1dXrF27NqZPnx5z586NrVu3no+SG1YzzJeRasY5k2VZLF68OG6++ea4/vrrh2zXbPPmTMelmebMzp0749JLL43W1tbo6OiIdevWRbFYHLRtM82XWsalmebLmjVr4uc//3l0dnaeUfvRmjMtNVV5DuVyuQH7WZadcmy49oMdT10t4zJ9+vSYPn16/357e3vs27cvvvGNb8Qtt9xyTutsdM0yX2rVjHPmwQcfjDfffDN++tOfDtu2mebNmY5LM82Z6dOnx44dO+I3v/lNdHV1xcKFC6O7u3vI0NIs86WWcWmW+bJv37545JFHYuPGjTF27Ngz7jcac6buK3J/8Ad/EBdddNEpq0wHDx48JamedNVVVw3avqWlJcaPH3/Oaj2fRjIug7npppvi3XffHe3yktIM82U0Xchz5qGHHor169fH5s2bY/Lkyadt20zzppZxGcyFOmcuvvjiuOaaa2L27NnR2dkZN9xwQ3z7298etG0zzZdaxmUwF+J8ef311+PgwYMxa9asaGlpiZaWluju7o5nnnkmWlpa4vjx46f0Ga05U/cgd/HFF8esWbNi06ZNA45v2rQp/uzP/mzQPu3t7ae037hxY8yePTvGjBlzzmo9n0YyLoPp6emJtra20S4vKc0wX0bThThnsiyLBx98MNauXRs//vGPY+rUqcP2aYZ5M5JxGcyFOGcGk2VZ9PX1DfpaM8yXoZxuXAZzIc6XuXPnxs6dO2PHjh392+zZs+Oee+6JHTt2xEUXXXRKn1GbMzXdGnGOrFmzJhszZky2atWq7O23384effTR7BOf+ET23nvvZVmWZU888UR277339rf/l3/5l+zjH/949thjj2Vvv/12tmrVqmzMmDHZP/7jP9brI5wTtY7Lt771rWzdunXZL37xi+yf//mfsyeeeCKLiKyrq6teH+GcOHz4cNbT05P19PRkEZF985vfzHp6erL3338/y7LmnS9ZVvvYNMuceeCBB7JCoZBt2bIlO3DgQP/229/+tr9NM86bkYxLs8yZpUuXZlu3bs327NmTvfnmm9myZcuyj33sY9nGjRuzLGvO+ZJltY9Ls8yXwXz0rtVzNWcaIshlWZaVy+XsD//wD7OLL744+5M/+ZMBt78vXLgwu/XWWwe037JlS3bjjTdmF198cXb11Vdny5cvP88Vnx+1jMtTTz2VfepTn8rGjh2bXX755dnNN9+cbdiwoQ5Vn1snb2f/6LZw4cIsy5p7vtQ6Ns0yZwYbk4jInn/++f42zThvRjIuzTJnvvSlL/X/7b3yyiuzuXPn9oeVLGvO+ZJltY9Ls8yXwXw0yJ2rOZPLsv//zToAAJJS9+/IAQAwMoIcAECiBDkAgEQJcgAAiRLkAAASJcgBACRKkAMASJQgBwCQKEEOACBRghwAQKIEOQCARAlyAACJ+n8gRERuXnL98AAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show(P)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ac0b81bb",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "SageMath 9.8",
   "language": "sage",
   "name": "sagemath-9.8"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.11.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}