{ "cells": [ { "cell_type": "code", "execution_count": 3, "id": "1494ba58-0b3b-4a26-8099-607d97f84dbb", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "Welcome to Nemo version 0.43.3\n", "\n", "Nemo comes with absolutely no warranty whatsoever\n", "\n", "Welcome to \n", "\n", "\u001b[31m _ _ _\u001b[39m\n", "\u001b[31m | | | | | |\u001b[39m\n", "\u001b[31m | |__| | ___ ___| | _____\u001b[39m\n", "\u001b[31m | __ |/ _ \\/ __| |/ / _ \\\u001b[39m\n", "\u001b[31m | | | | __/ (__| < __/\u001b[39m\n", "\u001b[31m |_| |_|\\___|\\___|_|\\_\\___|\u001b[39m\n", "\u001b[31m \u001b[39m\n", "Version\u001b[32m 0.30.12 \u001b[39m... \n", " ... which comes with absolutely no warranty whatsoever\n", "(c) 2015-2024 by Claus Fieker, Tommy Hofmann and Carlo Sircana\n", "\n", " ___ ____ ____ _ ____\n", " / _ \\ / ___| / ___| / \\ | _ \\ | Combining ANTIC, GAP, Polymake, Singular\n", "| | | |\\___ \\| | / _ \\ | |_) | | Type \"?Oscar\" for more information\n", "| |_| | ___) | |___ / ___ \\| _ < | Manual: https://docs.oscar-system.org\n", " \\___/ |____/ \\____/_/ \\_\\_| \\_\\ | Version 1.0.2\n", "\r" ] } ], "source": [ "using Oscar\n", "using LinearAlgebra\n", "using Oscar.Hecke" ] }, { "cell_type": "code", "execution_count": 4, "id": "d71b297d-bb23-4b48-979d-ff877145342c", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "twist (generic function with 1 method)" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "function twist(L, k)\n", " Lm = gram_matrix(L)\n", " n = minimum(size(Lm))\n", " Lm_twist = Lm * (k * identity_matrix(ZZ, n))\n", " L_twist = integer_lattice(gram = Lm_twist)\n", " return L_twist\n", "end" ] }, { "cell_type": "code", "execution_count": 7, "id": "dbf01d4d-0528-46bc-b50a-05190c74f721", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Integer lattice of rank 1 and degree 1\n", "with gram matrix\n", "[1]" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "Integer lattice of rank 1 and degree 1\n", "with gram matrix\n", "[2]" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "Integer lattice of rank 8 and degree 8\n", "with gram matrix\n", "[-2 1 0 0 0 0 0 0]\n", "[ 1 -2 1 0 0 0 0 0]\n", "[ 0 1 -2 1 0 0 0 1]\n", "[ 0 0 1 -2 1 0 0 0]\n", "[ 0 0 0 1 -2 1 0 0]\n", "[ 0 0 0 0 1 -2 1 0]\n", "[ 0 0 0 0 0 1 -2 0]\n", "[ 0 0 1 0 0 0 0 -2]" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "Integer lattice of rank 8 and degree 8\n", "with gram matrix\n", "[-4 2 0 0 0 0 0 0]\n", "[ 2 -4 2 0 0 0 0 0]\n", "[ 0 2 -4 2 0 0 0 2]\n", "[ 0 0 2 -4 2 0 0 0]\n", "[ 0 0 0 2 -4 2 0 0]\n", "[ 0 0 0 0 2 -4 2 0]\n", "[ 0 0 0 0 0 2 -4 0]\n", "[ 0 0 2 0 0 0 0 -4]" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "Integer lattice of rank 2 and degree 2\n", "with gram matrix\n", "[0 1]\n", "[1 0]" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "Integer lattice of rank 2 and degree 2\n", "with gram matrix\n", "[0 2]\n", "[2 0]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "Z1 = root_lattice(:I, 1)\n", "display(Z1)\n", "\n", "Z2 = twist(Z1, 2)\n", "display(Z2)\n", "\n", "E8 = twist( root_lattice(:E, 8), -1)\n", "display(E8)\n", "\n", "E82 = twist(E8, 2)\n", "display(E82)\n", "\n", "H = hyperbolic_plane_lattice()\n", "display(H)\n", "\n", "H2 = twist(H, 2)\n", "display(H2)" ] }, { "cell_type": "code", "execution_count": 8, "id": "ecc4d9e0-e377-4062-ac9d-48216d6391d4", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Integer lattice of rank 10 and degree 10\n", "with gram matrix\n", "[0 1 0 0 0 0 0 0 0 0]\n", "[1 0 0 0 0 0 0 0 0 0]\n", "[0 0 -2 1 0 0 0 0 0 0]\n", "[0 0 1 -2 1 0 0 0 0 0]\n", "[0 0 0 1 -2 1 0 0 0 1]\n", "[0 0 0 0 1 -2 1 0 0 0]\n", "[0 0 0 0 0 1 -2 1 0 0]\n", "[0 0 0 0 0 0 1 -2 1 0]\n", "[0 0 0 0 0 0 0 1 -2 0]\n", "[0 0 0 0 1 0 0 0 0 -2]" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "Integer lattice of rank 10 and degree 10\n", "with gram matrix\n", "[0 2 0 0 0 0 0 0 0 0]\n", "[2 0 0 0 0 0 0 0 0 0]\n", "[0 0 -4 2 0 0 0 0 0 0]\n", "[0 0 2 -4 2 0 0 0 0 0]\n", "[0 0 0 2 -4 2 0 0 0 2]\n", "[0 0 0 0 2 -4 2 0 0 0]\n", "[0 0 0 0 0 2 -4 2 0 0]\n", "[0 0 0 0 0 0 2 -4 2 0]\n", "[0 0 0 0 0 0 0 2 -4 0]\n", "[0 0 0 0 2 0 0 0 0 -4]" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "E10 = direct_sum(H, E8)[1]\n", "display(E10)\n", "\n", "E10_2 = twist(E10, 2)\n", "display(E10_2)" ] }, { "cell_type": "code", "execution_count": 9, "id": "d3969319-46b7-472b-8947-66d6184eed96", "metadata": {}, "outputs": [ { "ename": "LoadError", "evalue": "Type (:B) must be :A, :D, :E or :I", "output_type": "error", "traceback": [ "Type (:B) must be :A, :D, :E or :I", "", "Stacktrace:", " [1] root_lattice(R::Symbol, n::Int64)", " @ Hecke ~/.julia/packages/Hecke/rluKS/src/QuadForm/Quad/ZLattices.jl:686", " [2] top-level scope", " @ In[9]:1" ] } ], "source": [ "B7 = twist( root_lattice(:B, 7), -1)\n", "display(B7)" ] }, { "cell_type": "code", "execution_count": 11, "id": "9ab1a809-9b83-4905-87bd-4664a4291413", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Integer lattice of rank 8 and degree 8\n", "with gram matrix\n", "[-4 2 0 0 0 0 0 0]\n", "[ 2 -4 2 0 0 0 0 0]\n", "[ 0 2 -4 2 0 0 0 2]\n", "[ 0 0 2 -4 2 0 0 0]\n", "[ 0 0 0 2 -4 2 0 0]\n", "[ 0 0 0 0 2 -4 2 0]\n", "[ 0 0 0 0 0 2 -4 0]\n", "[ 0 0 2 0 0 0 0 -4]" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Sdp = H2\n", "Sen = direct_sum( twist(H, 2), twist(E8, 2) )[1]\n", "Lnikp = direct_sum(H, H, H, twist(E8, 2) )[1]\n", "\n", "Tdp = direct_sum(H, twist(H, 2), E8, E8)[1]\n", "Ten = direct_sum(H, twist(H, 2), twist(E8, 2) )[1]\n", "Lnikm = twist(E8, 2)" ] }, { "cell_type": "code", "execution_count": null, "id": "ad661ed1-2afe-4aee-88b8-024b634c481a", "metadata": {}, "outputs": [], "source": [ "# Return whether L is primary, that is whether L is integral and its discriminant group (see discriminant_group) is a p-group for some prime number p. \n", "# In case it is, p is also returned as second output.\n", "#Note that for unimodular lattices, this function returns (true, 1). \n", "# If the lattice is not primary, the second return value is -1 by default.\n", "\n", "display( is_primary_with_prime(H) ) #unimodular\n", "display( is_primary_with_prime(E8) ) #unimodular\n", "display( is_primary_with_prime(E10) ) #unimodular\n", "display( is_primary_with_prime(E10_2) ) #2-primary" ] }, { "cell_type": "code", "execution_count": null, "id": "2a2f7ae6-8f80-4af2-a083-ef124b254683", "metadata": {}, "outputs": [], "source": [ "display( is_elementary_with_prime(H) ) #unimodular\n", "display( is_elementary_with_prime(E8) ) #unimodular\n", "display( is_elementary_with_prime(E10) ) #unimodular\n", "display( is_elementary_with_prime(E10_2) ) #2-elementary" ] }, { "cell_type": "code", "execution_count": null, "id": "969e169d-f4c9-4fd6-96ab-ff9193a05dc0", "metadata": {}, "outputs": [], "source": [ "# Return the number of (positive, zero, negative) inertia of L.\n", "\n", "display(signature_tuple(H))\n", "display(signature_tuple(E8))\n", "display(signature_tuple(E10))\n", "display(signature_tuple(E10_2))" ] }, { "cell_type": "code", "execution_count": null, "id": "1c33c547-f54c-4433-be3b-125c52929dd4", "metadata": {}, "outputs": [], "source": [ "discriminant_group(E10)" ] }, { "cell_type": "code", "execution_count": null, "id": "34985de9-5577-4f3d-bff4-f5522136de75", "metadata": {}, "outputs": [], "source": [ "discriminant_group(E10_2)" ] }, { "cell_type": "code", "execution_count": null, "id": "e4161911-5402-4ff1-a269-9be43b6f060c", "metadata": {}, "outputs": [], "source": [ "L = direct_sum(Z2, H2, E82)[1]" ] }, { "cell_type": "code", "execution_count": null, "id": "eb8ccc7a-072e-422a-ba52-3237af518659", "metadata": {}, "outputs": [], "source": [ "display( signature_tuple(L) )\n", "display( is_primary_with_prime(L) )\n", "display( is_elementary_with_prime(L) )\n", "display( discriminant_group(L) )" ] }, { "cell_type": "markdown", "id": "ed290f9c-4a6b-4221-a4d0-d7d2a39dbf6f", "metadata": {}, "source": [ "# Coparity\n", "\n", "AE22: Compactifications of moduli spaces of K3 surfaces with a nonsymplectic involution\n", "\n", "**Definition 2.3.** We define an additional invariant, coparity $\\delta_H$ as follows: $\\delta=0$ if for all $x \\in A_H$ one has $q_H(x) \\equiv 0(\\bmod \\mathbb{Z})$ and $\\delta=1$ otherwise. We will call lattices with $\\delta_H=0$ co-even and lattices with $\\delta_H=1$ co-odd." ] }, { "cell_type": "code", "execution_count": 77, "id": "a5c5609c-140d-4960-af6b-3a6a777a120b", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "rad_invts (generic function with 1 method)" ] }, "execution_count": 77, "metadata": {}, "output_type": "execute_result" } ], "source": [ "function rad_invts(lat::ZZLat)\n", " display(\"------------------------------------\")\n", " is_elem = is_elementary_with_prime(lat)\n", " if is_elem[1] == false || is_elem[2] != 2\n", " display(\"Not a 2-elementary lattice:\")\n", " display(is_elem)\n", " return 0\n", " end\n", " #display(\"This is a p-elementary lattice.\")\n", "\n", " D_L = discriminant_group(lat)\n", " display(D_L)\n", " Q = D_L.gram_matrix_quadratic\n", " G = D_L.ab_grp\n", " #display(G)\n", "\n", " #display(\"Computing r..\")\n", " r = rank(lat)\n", "\n", " #display(\"Computing a..\")\n", " a = length( filter(x -> x == 2, elementary_divisors(G)) )\n", "\n", " #display(\"Computing delta...\")\n", " \n", " n = minimum(size(Q))\n", " diags = [ Q[i, i] for i in 1:n ]\n", " #show(\"Diagonal of Q:\")\n", " #show(diags)\n", " are_diags_ints = map(is_integer, diags)\n", " all_integer_diags = reduce(&, are_diags_ints)\n", " # = 1 if all integers, = 0 if any non-integer \n", " \n", " delta = 1 - all_integer_diags\n", " # delta = 0 <=> image in Z, dellta=1 <=> non-integral image\n", "\n", " display(\"------------------------------------\")\n", " \n", " return (r, a, delta)\n", "end" ] }, { "cell_type": "code", "execution_count": 78, "id": "a2eb443b-5ff4-4c4b-9746-6a3a7c568811", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\"------------------------------------\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "Finite quadratic module\n", " over integer ring\n", "Abelian group: (Z/2)^2\n", "Bilinear value module: Q/Z\n", "Quadratic value module: Q/2Z\n", "Gram matrix quadratic form: \n", "[ 0 1//2]\n", "[1//2 0]" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "\"------------------------------------\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "(2, 2, 0)" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "\"------------------------------------\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "Finite quadratic module\n", " over integer ring\n", "Abelian group: (Z/2)^10\n", "Bilinear value module: Q/Z\n", "Quadratic value module: Q/2Z\n", "Gram matrix quadratic form: \n", "[ 0 1//2 0 0 0 0 0 0 0 0]\n", "[1//2 0 0 0 0 0 0 0 0 0]\n", "[ 0 0 0 1//2 0 0 0 0 0 1//2]\n", "[ 0 0 1//2 1 0 0 0 0 0 0]\n", "[ 0 0 0 0 1 0 0 0 0 1//2]\n", "[ 0 0 0 0 0 0 1//2 0 1//2 0]\n", "[ 0 0 0 0 0 1//2 0 0 0 1//2]\n", "[ 0 0 0 0 0 0 0 1 1//2 0]\n", "[ 0 0 0 0 0 1//2 0 1//2 1 1//2]\n", "[ 0 0 1//2 0 1//2 0 1//2 0 1//2 0]" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "\"------------------------------------\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "(10, 10, 0)" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "\"------------------------------------\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "Finite quadratic module\n", " over integer ring\n", "Abelian group: (Z/2)^8\n", "Bilinear value module: Q/Z\n", "Quadratic value module: Q/2Z\n", "Gram matrix quadratic form: \n", "[ 0 1//2 0 0 0 0 0 1//2]\n", "[1//2 1 0 0 0 0 0 0]\n", "[ 0 0 1 0 0 0 0 1//2]\n", "[ 0 0 0 0 1//2 0 1//2 0]\n", "[ 0 0 0 1//2 0 0 0 1//2]\n", "[ 0 0 0 0 0 1 1//2 0]\n", "[ 0 0 0 1//2 0 1//2 1 1//2]\n", "[1//2 0 1//2 0 1//2 0 1//2 0]" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "\"------------------------------------\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "(14, 8, 0)" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "\"------------------------------------\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "Finite quadratic module\n", " over integer ring\n", "Abelian group: (Z/2)^2\n", "Bilinear value module: Q/Z\n", "Quadratic value module: Q/2Z\n", "Gram matrix quadratic form: \n", "[ 0 1//2]\n", "[1//2 0]" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "\"------------------------------------\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "(20, 2, 0)" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "\"------------------------------------\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "Finite quadratic module\n", " over integer ring\n", "Abelian group: (Z/2)^10\n", "Bilinear value module: Q/Z\n", "Quadratic value module: Q/2Z\n", "Gram matrix quadratic form: \n", "[ 0 1//2 0 0 0 0 0 0 0 0]\n", "[1//2 0 0 0 0 0 0 0 0 0]\n", "[ 0 0 0 1//2 0 0 0 0 0 1//2]\n", "[ 0 0 1//2 1 0 0 0 0 0 0]\n", "[ 0 0 0 0 1 0 0 0 0 1//2]\n", "[ 0 0 0 0 0 0 1//2 0 1//2 0]\n", "[ 0 0 0 0 0 1//2 0 0 0 1//2]\n", "[ 0 0 0 0 0 0 0 1 1//2 0]\n", "[ 0 0 0 0 0 1//2 0 1//2 1 1//2]\n", "[ 0 0 1//2 0 1//2 0 1//2 0 1//2 0]" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "\"------------------------------------\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "(12, 10, 0)" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "\"------------------------------------\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "Finite quadratic module\n", " over integer ring\n", "Abelian group: (Z/2)^8\n", "Bilinear value module: Q/Z\n", "Quadratic value module: Q/2Z\n", "Gram matrix quadratic form: \n", "[ 0 1//2 0 0 0 0 0 1//2]\n", "[1//2 1 0 0 0 0 0 0]\n", "[ 0 0 1 0 0 0 0 1//2]\n", "[ 0 0 0 0 1//2 0 1//2 0]\n", "[ 0 0 0 1//2 0 0 0 1//2]\n", "[ 0 0 0 0 0 1 1//2 0]\n", "[ 0 0 0 1//2 0 1//2 1 1//2]\n", "[1//2 0 1//2 0 1//2 0 1//2 0]" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "\"------------------------------------\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "(8, 8, 0)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "display( rad_invts(Sdp) )\n", "display( rad_invts( Sen ) )\n", "display( rad_invts( Lnikp ) )\n", "\n", "display( rad_invts( Tdp ) )\n", "display( rad_invts( Ten ) )\n", "display( rad_invts( Lnikm ) )" ] }, { "cell_type": "code", "execution_count": 43, "id": "92c284c1-d5c2-449c-820f-a6fe22916d52", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(true, 2)" ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "lat = Lnikp\n", "is_elem = is_elementary_with_prime(lat)\n", "if is_elem[1] == false || is_elem[2] != 2\n", " display(\"Not a 2-elementary lattice:\")\n", " display(is_elem)\n", " return 0\n", "end\n", "is_elem" ] }, { "cell_type": "code", "execution_count": 53, "id": "1e772ed7-fa85-4406-b412-0e233a4fe86d", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Finite quadratic module\n", " over integer ring\n", "Abelian group: (Z/2)^8\n", "Bilinear value module: Q/Z\n", "Quadratic value module: Q/2Z\n", "Gram matrix quadratic form: \n", "[ 0 1//2 0 0 0 0 0 1//2]\n", "[1//2 1 0 0 0 0 0 0]\n", "[ 0 0 1 0 0 0 0 1//2]\n", "[ 0 0 0 0 1//2 0 1//2 0]\n", "[ 0 0 0 1//2 0 0 0 1//2]\n", "[ 0 0 0 0 0 1 1//2 0]\n", "[ 0 0 0 1//2 0 1//2 1 1//2]\n", "[1//2 0 1//2 0 1//2 0 1//2 0]" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "(Z/2)^8" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "D_L = discriminant_group(lat);\n", "display(D_L)\n", "G = D_L.ab_grp;\n", "display(G)\n", "Q = D_L.gram_matrix_quadratic;" ] }, { "cell_type": "code", "execution_count": 35, "id": "277d4964-35bb-4295-ab56-e1fee52bd5b4", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "2" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "D_L = discriminant_group(lat)\n", "Q = D_L.gram_matrix_quadratic\n", "G = D_L.ab_grp\n", "\n", "# Computing r\n", "r = rank(lat)" ] }, { "cell_type": "code", "execution_count": null, "id": "d482d7b7-f2fe-4e83-b376-ec8293105c22", "metadata": {}, "outputs": [], "source": [ "# Computing a\n", " a = length( filter(x -> x == 2, elementary_divisors(G)) )" ] }, { "cell_type": "code", "execution_count": null, "id": "bce8b460-bef0-40bd-8767-fdd79fe55717", "metadata": {}, "outputs": [], "source": [ "n = minimum(size(Q))\n", "diags = [ Q[i, i] for i in 1:n ]\n", "#show(\"Diagonal of Q:\")\n", "#show(diags)\n", "are_diags_ints = map(is_integer, diags)\n", "all_integer_diags = reduce(&, are_diags_ints)\n", "# = 1 if all integers, = 0 if any non-integer " ] }, { "cell_type": "code", "execution_count": null, "id": "e4704912-cc55-4543-a26a-2b827cde6f94", "metadata": {}, "outputs": [], "source": [ "Q" ] }, { "cell_type": "code", "execution_count": null, "id": "d40a6e0d-718d-4623-b609-57bf2edc4337", "metadata": {}, "outputs": [], "source": [ "delta = 1 - all_integer_diags\n", " # delta = 0 <=> image in Z, dellta=1 <=> non-integral image" ] }, { "cell_type": "code", "execution_count": null, "id": "59a9a910-b644-4fd1-9715-ee18815619fe", "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "id": "2ca6e741-8e0c-4786-85aa-0f4f5fe00259", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Julia 1.10.3", "language": "julia", "name": "julia-1.10" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.10.3" } }, "nbformat": 4, "nbformat_minor": 5 }