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script/research/ec/valuation.ipynb

@@ -551,7 +551,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 22,
+   "execution_count": 13,
    "id": "ce94255a",
    "metadata": {},
    "outputs": [
@@ -561,7 +561,7 @@
        "(True, False)"
       ]
      },
-     "execution_count": 22,
+     "execution_count": 13,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -570,15 +570,85 @@
     "R.<x> = FunctionField(GF(11)); _.<Y> = R[]\n",
     "K.<y> = R.extension(Y^2 - x^3 - 4*x)\n",
     "P = (2, 4)\n",
-    "I = K.maximal_order().ideal(x - 2, y - 4)\n",
-    "# (x - 2) in I^2 should be false if I = <x - 2>\n",
-    "x - 2 in I, x - 2 in I^2\n",
-    "# But for some reason I need y - 4 as well..."
+    "o = R.maximal_order()\n",
+    "O = K.maximal_order()\n",
+    "M = o.ideal(x - 2)\n",
+    "# if I = <x - 2> then (x - 2) in I^2 is false \n",
+    "x - 2 in M, x - 2 in M^2"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "052087b6",
+   "metadata": {},
+   "source": [
+    "Also verify that the point is ordinary (nonsingular)"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 17,
+   "execution_count": 8,
+   "id": "1a218ce1",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "True"
+      ]
+     },
+     "execution_count": 8,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "M.is_prime()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "id": "e737ac7f",
+   "metadata": {
+    "scrolled": true
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[(Ideal (x + 9, y + 4) of Maximal order of Function field in y defined by y^2 + 10*x^3 + 7*x,\n",
+       "  1,\n",
+       "  1),\n",
+       " (Ideal (x + 9, y + 7) of Maximal order of Function field in y defined by y^2 + 10*x^3 + 7*x,\n",
+       "  1,\n",
+       "  1)]"
+      ]
+     },
+     "execution_count": 14,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "O.decomposition(M)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "id": "867a9f68",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# I can't seem to use M for f\n",
+    "# This is a workaround until I find a fix\n",
+    "M = O.ideal(x - 2, y - 4)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
    "id": "2a10bcd6",
    "metadata": {},
    "outputs": [
@@ -588,19 +658,19 @@
        "(True, True, False)"
       ]
      },
-     "execution_count": 17,
+     "execution_count": 20,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
     "f = y - 2*x\n",
-    "f in I, f in I^2, f in I^3"
+    "f in M, f in M^2, f in M^3"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 20,
+   "execution_count": 21,
    "id": "73406fd6",
    "metadata": {},
    "outputs": [
@@ -610,13 +680,13 @@
        "2"
       ]
      },
-     "execution_count": 20,
+     "execution_count": 21,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "d = [i for i in range(1, 10) if f in I^i]\n",
+    "d = [i for i in range(1, 10) if f in M^i]\n",
     "max(d)"
    ]
   }