diff --git a/script/research/ec/valuation.ipynb b/script/research/ec/valuation.ipynb index bbcbef553..3f41d409d 100644 --- a/script/research/ec/valuation.ipynb +++ b/script/research/ec/valuation.ipynb @@ -734,11 +734,107 @@ "source": [ "As observed we have a uniformizer $t$ with $k = 2$." ] + }, + { + "cell_type": "markdown", + "id": "a05bc126", + "metadata": {}, + "source": [ + "# Valuation with Places\n", + "\n", + "Using sage valuation rings since local ring is a DVR" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "3da404fc", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[Place (x, y),\n", + " Place (x + 3, y + 4),\n", + " Place (x + 3, y + 7),\n", + " Place (x + 5, y + 3),\n", + " Place (x + 5, y + 8),\n", + " Place (x + 7, y + 5),\n", + " Place (x + 7, y + 6),\n", + " Place (x + 9, y + 4),\n", + " Place (x + 9, y + 7),\n", + " Place (x + 10, y + 4),\n", + " Place (x + 10, y + 7)]" + ] + }, + "execution_count": 2, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "K. = FunctionField(GF(11))\n", + "_. = K[]\n", + "L. = K.extension(Y^2 - x^3 - 4*x)\n", + "L.places_finite()" + ] + }, + { + "cell_type": "markdown", + "id": "e4719879", + "metadata": {}, + "source": [ + "We are interested in $P = (2, 4)$ so we want the place $(x - 2, x - 4) = (x + 9, y + 7)$." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "f87c8b23", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "Place (x + 9, y + 7)" + ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# I don't know how to actually construct this yet\n", + "p = L.places_finite()[-3]\n", + "p" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "3d007b0b", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "2" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(y - 2*x).valuation(p)" + ] } ], "metadata": { "kernelspec": { - "display_name": "SageMath 9.6", + "display_name": "SageMath 9.5", "language": "sage", "name": "sagemath" }, @@ -752,7 +848,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.10.5" + "version": "3.10.7" } }, "nbformat": 4,