quadratic_spaces.ipynb

{
 "cells": [
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     "text": [
      "\n",
      "Welcome to Nemo version 0.16.4-dev\n",
      "\n",
      "Nemo comes with absolutely no warranty whatsoever\n",
      "\n",
      "\n",
      "Welcome to \n"
     ]
    },
    {
     "data": {
      "text/html": [
       "$\\require{action}$"
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     "text": [
      "\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 0.7.3-dev \u001b[39m... \n",
      " ... which comes with absolutely no warranty whatsoever\n",
      "(c) 2015-2019 by Claus Fieker, Tommy Hofmann and Carlo Sircana\n",
      "\n"
     ]
    }
   ],
   "source": [
    "using Hecke;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "($Qx$, $x$)"
      ],
      "text/plain": [
       "(Univariate Polynomial Ring in x over Rational Field, x)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Qx, x = PolynomialRing(FlintQQ, \"x\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "($K$, $a$)"
      ],
      "text/plain": [
       "(Number field over Rational Field with defining polynomial x^2-5, a)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K, a = NumberField(x^2 - 5, \"a\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{bmatrix}382a+2054&70a+350&4&8&8a+8\\\\70a+350&14a+70&0&0&0\\\\4&0&4&0&0\\\\8&0&0&4&0\\\\8a+8&0&0&0&2a+2\\\\\\end{bmatrix}$"
      ],
      "text/plain": [
       "[382*a+2054  70*a+350  4  8  8*a+8]\n",
       "[  70*a+350   14*a+70  0  0      0]\n",
       "[         4         0  4  0      0]\n",
       "[         8         0  0  4      0]\n",
       "[     8*a+8         0  0  0  2*a+2]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G = matrix(K, 5, 5,[382*a+2054, 70*a+350, 4, 8, 8*a+8, 70*a+350, 14*a+70, 0, 0, 0, 4, 0, 4, 0, 0, 8, 0, 0, 4, 0, 8*a+8, 0, 0, 0, 2*a+2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Quadratic space over\n",
       "Number field over Rational Field with defining polynomial x^2-5\n",
       "with Gram matrix\n",
       "[382*a+2054  70*a+350  4  8  8*a+8]\n",
       "[  70*a+350   14*a+70  0  0      0]\n",
       "[         4         0  4  0      0]\n",
       "[         8         0  0  4      0]\n",
       "[     8*a+8         0  0  0  2*a+2]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V = quadratic_space(K, G)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "OK = maximal_order(K);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$-1$"
      ],
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hasse_invariant(V, 2 * OK)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hasse_invariant(V, real_places(K)[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "false"
      ],
      "text/plain": [
       "false"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "isequivalent(V, quadratic_space(K, 5), real_places(K)[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "false"
      ],
      "text/plain": [
       "false"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "isequivalent(V, quadratic_space(K, 5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(677376*a+1128960, Dict{NfAbsOrdIdl{AnticNumberField,nf_elem},Int64}(<2, 2>\n",
       "Norm: 4\n",
       "Minimum: 2\n",
       "basis_matrix \n",
       "[2  0]\n",
       "[0  2]\n",
       "two normal wrt: 2 => -1,<7, 7>\n",
       "Norm: 49\n",
       "Minimum: 7\n",
       "principal generator 7\n",
       "basis_matrix \n",
       "[7  0]\n",
       "[0  7]\n",
       "two normal wrt: 7 => -1), Tuple{InfPlc,Int64}[(Real place of \n",
       "Number field over Rational Field with defining polynomial x^2-5\n",
       "corresponding to the root [-2.2360679774997896964091736687312762354 +/- 6.13e-38] + i*0, 1), (Real place of \n",
       "Number field over Rational Field with defining polynomial x^2-5\n",
       "corresponding to the root [2.2360679774997896964091736687312762354 +/- 6.13e-38] + i*0, 0)])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "invariants(V)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{verbatim}\n",
       "invariants(M::QuadSpace)\n",
       "      -> FieldElem, Dict{NfOrdIdl, Int}, Vector{Tuple{InfPlc, Int}}\n",
       "\\end{verbatim}\n",
       "Returns a triple \\texttt{(d, H, I)} of invariants of \\texttt{M}, which determine the equivalence class completely. The element \\texttt{d} is the determinant of a Gram matrix, \\texttt{H} contains the non-trivial Hasse invariants and \\texttt{I} contains for each real place the negative index of inertia.\n",
       "\n",
       "Note that \\texttt{d} is determined only modulo squares.\n",
       "\n"
      ],
      "text/markdown": [
       "```\n",
       "invariants(M::QuadSpace)\n",
       "      -> FieldElem, Dict{NfOrdIdl, Int}, Vector{Tuple{InfPlc, Int}}\n",
       "```\n",
       "\n",
       "Returns a triple `(d, H, I)` of invariants of `M`, which determine the equivalence class completely. The element `d` is the determinant of a Gram matrix, `H` contains the non-trivial Hasse invariants and `I` contains for each real place the negative index of inertia.\n",
       "\n",
       "Note that `d` is determined only modulo squares.\n"
      ],
      "text/plain": [
       "\u001b[36m  invariants(M::QuadSpace)\u001b[39m\n",
       "\u001b[36m        -> FieldElem, Dict{NfOrdIdl, Int}, Vector{Tuple{InfPlc, Int}}\u001b[39m\n",
       "\n",
       "  Returns a triple \u001b[36m(d, H, I)\u001b[39m of invariants of \u001b[36mM\u001b[39m, which determine the\n",
       "  equivalence class completely. The element \u001b[36md\u001b[39m is the determinant of a Gram\n",
       "  matrix, \u001b[36mH\u001b[39m contains the non-trivial Hasse invariants and \u001b[36mI\u001b[39m contains for each\n",
       "  real place the negative index of inertia.\n",
       "\n",
       "  Note that \u001b[36md\u001b[39m is determined only modulo squares."
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "?invariants(V)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1, Dict{NfAbsOrdIdl{AnticNumberField,nf_elem},Int64}(), Tuple{InfPlc,Int64}[(Real place of \n",
       "Number field over Rational Field with defining polynomial x^2-5\n",
       "corresponding to the root [-2.2360679774997896964091736687312762354 +/- 6.13e-38] + i*0, 0), (Real place of \n",
       "Number field over Rational Field with defining polynomial x^2-5\n",
       "corresponding to the root [2.2360679774997896964091736687312762354 +/- 6.13e-38] + i*0, 0)])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "invariants(quadratic_space(K, 5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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