{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Hilibert Class Polynomial\n",
    "Yasuaki Honda at gmail dot com\n",
    "\n",
    "Hilbert Class Polynomial \\\\(P_D(x)\\\\)of determinant \\\\(D\\\\) is defined as follows:\n",
    "$$ P_{D}(x)=\\prod_{i=1}^{h_{D}}{\\left(x-j\\left(a_{i}\\right)\\right)} $$\n",
    "where \\\\(h_D\\\\) is the class number of the determinant \\\\(D\\\\).\n",
    "\n",
    "The fact that \\\\(P_D(x) \\in Z[x]\\\\), meaning that all the coefficients of \\\\(P_D(x)\\\\) are integers. This fact can be used to determine the exact value of \\\\(j(x)\\\\) for \\\\(x\\\\) a quadratic irrational.\n",
    "\n",
    "Note that all the tools to compute the Hilbert Class Polynomial is already defined:\n",
    "\n",
    "- \\\\(D\\\\) can be computed using **FindDetIrr(p)** where p is a quadratic irrational.\n",
    "- \\\\(a_{i}\\\\) can be computed using **FindReducedQuadIrrD(D)**.\n",
    "- \\\\(h_D\\\\) can be computed using **FindClassNumberD(D)**.\n",
    "- \\\\(j(x)\\\\) can be computed numerically using **float_modular_j(x)** or **bfloat_modular_j(x)**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "install_github(\"YasuakiHonda\",\"modular_j\",\"main\")$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "; processing (IN-PACKAGE :MAXIMA)\n",
      "; processing (LET NIL ...)\n",
      "; processing (DEFPROP |$InFD| ...)\n",
      "; processing (ADD2LNC (QUOTE |$InFD|) ...)\n",
      "; processing (DEFMTRFUN (|$InFD| $ANY ...) ...)\n",
      "; processing (DEFPROP |$Ta| ...)\n",
      "; processing (ADD2LNC (QUOTE |$Ta|) ...)\n",
      "; processing (DEFMTRFUN (|$Ta| $ANY ...) ...)\n",
      "; processing (DEFPROP |$FindInFD| ...)\n",
      "; processing (ADD2LNC (QUOTE |$FindInFD|) ...)\n",
      "; processing (DEFMTRFUN (|$FindInFD| $ANY ...) ...)\n",
      "; processing (LET NIL ...)\n",
      "; processing (DEFPROP $FLOAT_MODULAR_J ...)\n",
      "; processing (ADD2LNC (QUOTE $FLOAT_MODULAR_J) ...)\n",
      "; processing (DEFMTRFUN ($FLOAT_MODULAR_J $ANY ...) ...)\n",
      "; processing (DEFPROP $BFLOAT_MODULAR_J ...)\n",
      "; processing (ADD2LNC (QUOTE $BFLOAT_MODULAR_J) ...)\n",
      "; processing (DEFMTRFUN ($BFLOAT_MODULAR_J $ANY ...) ...)\n",
      "; processing (DEFPROP |$Checkabc| ...)\n",
      "; processing (ADD2LNC (QUOTE |$Checkabc|) ...)\n",
      "; processing (DEFMTRFUN (|$Checkabc| $ANY ...) ...)\n",
      "; processing (DEFPROP |$FindReducedQuadIrrD| ...)\n",
      "; processing (ADD2LNC (QUOTE |$FindReducedQuadIrrD|) ...)\n",
      "; processing (DEFMTRFUN (|$FindReducedQuadIrrD| $ANY ...) ...)\n",
      "; processing (DEFPROP |$FindClassNumberD| ...)\n",
      "; processing (ADD2LNC (QUOTE |$FindClassNumberD|) ...)\n",
      "; processing (DEFMTRFUN (|$FindClassNumberD| $FIXNUM ...) ...)\n",
      "; processing (DEFPROP |$FindClassPolyD| ...)\n",
      "; processing (ADD2LNC (QUOTE |$FindClassPolyD|) ...)\n",
      "; processing (DEFMTRFUN (|$FindClassPolyD| $ANY ...) ...)\n",
      "; processing (DEFPROP $IRR_NUMBER ...)\n",
      "; processing (ADD2LNC (QUOTE $IRR_NUMBER) ...)\n",
      "; processing (DEFMTRFUN ($IRR_NUMBER $ANY ...) ...)\n",
      "; processing (DEFPROP |$FindDetIrr| ...)\n",
      "; processing (ADD2LNC (QUOTE |$FindDetIrr|) ...)\n",
      "; processing (DEFMTRFUN (|$FindDetIrr| $ANY ...) ...)\n",
      "; processing (DEFPROP $EXACT_MODULAR_J ...)\n",
      "; processing (ADD2LNC (QUOTE $EXACT_MODULAR_J) ...)\n",
      "; processing (DEFMTRFUN ($EXACT_MODULAR_J $ANY ...) ...)[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::|$InFD-IMPL| in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::|$InFD| in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::|$Ta-IMPL| in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::|$Ta| in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::|$FindInFD-IMPL| in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::|$FindInFD| in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::$FLOAT_MODULAR_J-IMPL in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::$FLOAT_MODULAR_J in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::$BFLOAT_MODULAR_J-IMPL in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::$BFLOAT_MODULAR_J in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::|$Checkabc-IMPL| in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::|$Checkabc| in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::|$FindReducedQuadIrrD-IMPL| in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::|$FindReducedQuadIrrD| in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::|$FindClassNumberD-IMPL| in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::|$FindClassNumberD| in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::|$FindClassPolyD-IMPL| in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::|$FindClassPolyD| in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::$IRR_NUMBER-IMPL in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::$IRR_NUMBER in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::|$FindDetIrr-IMPL| in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::|$FindDetIrr| in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::$EXACT_MODULAR_J-IMPL in DEFUN\n",
      "[SB-KERNEL:REDEFINITION-WITH-DEFUN] redefining MAXIMA::$EXACT_MODULAR_J in DEFUN\n"
     ]
    }
   ],
   "source": [
    "asdf_load(\"modular_j\")$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{3}$}P_{D}(x)=\\prod_{i=1}^{h_{D}}{\\left(x-j\\left(a_{i}\\right)\\right)}\\]"
      ],
      "text/plain": [
       "                                    h\n",
       "                                     D\n",
       "                                   /===\\\n",
       "                                    ! !\n",
       "(%o3)                      P (x) =  ! !  (x - j(a ))\n",
       "                            D       ! !          i\n",
       "                                   i = 1"
      ],
      "text/x-maxima": [
       "P[D](x) = 'product(x-j(a[i]),i,1,h[D])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P[D](x)=product(x-j(a[i]),i,1,h[D]);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{4}$}\\left[ \\frac{\\sqrt{15}\\,i-1}{4} , \\frac{\\sqrt{15}\\,i-1}{2} \\right] \\]"
      ],
      "text/plain": [
       "                       sqrt(15) %i - 1  sqrt(15) %i - 1\n",
       "(%o4)                 [---------------, ---------------]\n",
       "                              4                2"
      ],
      "text/x-maxima": [
       "[(sqrt(15)*%i-1)/4,(sqrt(15)*%i-1)/2]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a:FindReducedQuadIrrD(-15);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{5}$}2\\]"
      ],
      "text/plain": [
       "(%o5)                                  2"
      ],
      "text/x-maxima": [
       "2"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FindClassNumberD(-15);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{6}$}\\left(x+191657.8328625472\\right)\\,\\left(x-1.70530256582424 \\times 10^{-13}\\,i-632.8328625472075\\right)\\]"
      ],
      "text/plain": [
       "(%o6) (x + 191657.8328625472) (x - 1.70530256582424e-13 %i - 632.8328625472075)"
      ],
      "text/x-maxima": [
       "(x+191657.8328625472)*(x-1.70530256582424e-13*%i-632.8328625472075)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "product(x-float_modular_j(a[i]),i,1,2);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{7}$}x^2-1.70530256582424 \\times 10^{-13}\\,i\\,x+191025.0\\,x-3.268345941408153 \\times 10^{-8}\\,i-1.21287375 \\times 10^{+8}\\]"
      ],
      "text/plain": [
       "       2\n",
       "(%o7) x  - 1.70530256582424e-13 %i x + 191025.0 x - 3.268345941408153e-8 %i\n",
       "                                                                - 1.21287375e+8"
      ],
      "text/x-maxima": [
       "x^2-1.70530256582424e-13*%i*x+191025.0*x-3.268345941408153e-8*%i-1.21287375e+8"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "aproxpoly:expand(%);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{8}$}191025\\]"
      ],
      "text/plain": [
       "(%o8)                               191025"
      ],
      "text/x-maxima": [
       "191025"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(cround(z):=round(realpart(z))+round(imagpart(z)), cround(coeff(aproxpoly,x,1)));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{9}$}-121287375\\]"
      ],
      "text/plain": [
       "(%o9)                             - 121287375"
      ],
      "text/x-maxima": [
       "-121287375"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cround(coeff(aproxpoly,x,0));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{10}$}x^2+191025\\,x-121287375\\]"
      ],
      "text/plain": [
       "                            2\n",
       "(%o10)                     x  + 191025 x - 121287375"
      ],
      "text/x-maxima": [
       "x^2+191025*x-121287375"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x^2+%th(2)*x+%th(1);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{11}$}x^2+191025\\,x-121287375\\]"
      ],
      "text/plain": [
       "                            2\n",
       "(%o11)                     x  + 191025 x - 121287375"
      ],
      "text/x-maxima": [
       "x^2+191025*x-121287375"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FindClassPolyD(x,-15);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Maxima",
   "language": "maxima",
   "name": "maxima"
  },
  "language_info": {
   "codemirror_mode": "maxima",
   "file_extension": ".mac",
   "mimetype": "text/x-maxima",
   "name": "maxima",
   "pygments_lexer": "maxima",
   "version": "5.44.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}