{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Computing value of j function for imaginary quadratic irrational numbers\n",
    "\n",
    "### Yasuaki.Honda at gmail.com\n",
    "\n",
    "Klein's elliptic modular j-invariant, or j function in short, has a notable property that the special value for the imaginary quadratic irrational numbers can be computed exactly. The following is the algorithm implemented in Maxima in this notebook.\n",
    "\n",
    "- Let's assume \\\\(t\\\\) is an imaginary quadratic irrational number, the argument for the j function.\n",
    "- Compute the determinant \\\\(D\\\\) of \\\\(t\\\\).\n",
    "- List up all the reduced imaginary quadratic irrational numbers \\\\(x_0,\\cdots, x_n \\\\)whose determinant is \\\\(D\\\\).\n",
    "- Note that the number of such irrationals \\\\(n\\\\) is called the class number of the determinant \\\\(D\\\\).\n",
    "- Compute the aproximate class equation \\\\( (x-j(x_0))\\cdots (x-j(x_n))\\\\) where \\\\(j(x_i)\\\\) is computed using q-expansion of j function.\n",
    "- Find the exact class equation by rounding up the coefficients of the aproximate class equation and solve it using solve() algebraically.\n",
    "- Find the correspondance between aproximate \\\\(j(x_i)\\\\) and the exact \\\\(j(x_i)\\\\) to know the exact value of \\\\(j(x_i)\\\\)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Manual\n",
    "**InFD(p)** p is a complex number. `InFD(p)` checks if p is in the fundamental domain of \\\\(j(z)\\\\).\n",
    "\n",
    "**FindInFD(p)** p is a complex number. `FindInFD(p)` returns a complex number in the fundamental domain and equivalent to p. It shows the necessary transformations, too.\n",
    "\n",
    "**float_modular_j(t)** t is a complex float number. `float_modular_j(t)` numerically computes the value of \\\\(j(t)\\\\).\n",
    "\n",
    "**FindReducedQuadIrrD(D)** D is an integer. `FindReducedQuadIrrD(D)` computes the list of all the reduced imaginary quadratic irrational numbers whose determinant is \\\\(D\\\\).\n",
    "\n",
    "**FindClassNumberD(D)** D is an integer. `FindClassNumberD(D)` computes the class number of the determinant \\\\(D\\\\).\n",
    "\n",
    "**FindClassPolyD(var,D)** var is a variable, D is an integer. `FindClassPolyD(var,D)` computes the Hilbert class polynomial of variable \\\\(var\\\\), the determinant equal to \\\\(D\\\\).\n",
    "\n",
    "**exact_modular_j(t)** t is an imaginary quadratic irrational number of the form \\\\( \\frac{a_1+a_2\\,\\sqrt{a_3}}{a_4} \\\\) where \\\\(a_1,\\cdots,a_4\\\\) are integers, \\\\(a_2, a_4\\\\) positive and \\\\(a_3\\\\) negative. `exact_modular_j(t)` computes the exact value (which is an algebraic number) of \\\\(j(t)\\\\) using the above algorithm."
   ]
  },
  {
   "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": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{3}$}\\frac{\\sqrt{3}\\,i+1}{2}\\]"
      ],
      "text/plain": [
       "                                sqrt(3) %i + 1\n",
       "(%o3)                           --------------\n",
       "                                      2"
      ],
      "text/x-maxima": [
       "(sqrt(3)*%i+1)/2"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p:(1+sqrt(-3))/2;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{4}$}-1.989519660128281 \\times 10^{-13}\\]"
      ],
      "text/plain": [
       "(%o4)                       - 1.989519660128281e-13"
      ],
      "text/x-maxima": [
       "-1.989519660128281e-13"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "float_modular_j(p);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{5}$}0\\]"
      ],
      "text/plain": [
       "(%o5)                                  0"
      ],
      "text/x-maxima": [
       "0"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exact_modular_j(p);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{6}$}\\frac{\\sqrt{15}\\,i+1}{2}\\]"
      ],
      "text/plain": [
       "                                sqrt(15) %i + 1\n",
       "(%o6)                           ---------------\n",
       "                                       2"
      ],
      "text/x-maxima": [
       "(sqrt(15)*%i+1)/2"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p:(1+sqrt(-15))/2;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{7}$}-\\frac{135\\,\\left(637\\,\\sqrt{5}+1415\\right)}{2}\\]"
      ],
      "text/plain": [
       "                            135 (637 sqrt(5) + 1415)\n",
       "(%o7)                     - ------------------------\n",
       "                                       2"
      ],
      "text/x-maxima": [
       "-(135*(637*sqrt(5)+1415))/2"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exact_modular_j(p),factor;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{8}$}-191657.8328625472\\]"
      ],
      "text/plain": [
       "(%o8)                         - 191657.8328625472"
      ],
      "text/x-maxima": [
       "-191657.8328625472"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%,numer;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{9}$}-191657.8328625472\\]"
      ],
      "text/plain": [
       "(%o9)                         - 191657.8328625472"
      ],
      "text/x-maxima": [
       "-191657.8328625472"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "float_modular_j(p);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{10}$}\\frac{\\sqrt{23}\\,i-1}{2}\\]"
      ],
      "text/plain": [
       "                                sqrt(23) %i - 1\n",
       "(%o10)                          ---------------\n",
       "                                       2"
      ],
      "text/x-maxima": [
       "(sqrt(23)*%i-1)/2"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p:(-1+sqrt(-23))/2;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{11}$}-23\\]"
      ],
      "text/plain": [
       "(%o11)                               - 23"
      ],
      "text/x-maxima": [
       "-23"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D:FindDetIrr(p);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{12}$}\\left[ \\frac{\\sqrt{23}\\,i-1}{4} , \\frac{\\sqrt{23}\\,i+1}{4} , \\frac{\\sqrt{23}\\,i-1}{2} \\right] \\]"
      ],
      "text/plain": [
       "               sqrt(23) %i - 1  sqrt(23) %i + 1  sqrt(23) %i - 1\n",
       "(%o12)        [---------------, ---------------, ---------------]\n",
       "                      4                4                2"
      ],
      "text/x-maxima": [
       "[(sqrt(23)*%i-1)/4,(sqrt(23)*%i+1)/4,(sqrt(23)*%i-1)/2]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FindReducedQuadIrrD(D);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{13}$}3\\]"
      ],
      "text/plain": [
       "(%o13)                                 3"
      ],
      "text/x-maxima": [
       "3"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FindClassNumberD(D);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{14}$}x^3+3491750\\,x^2-5151296875\\,x+12771880859375\\]"
      ],
      "text/plain": [
       "                 3            2\n",
       "(%o14)          x  + 3491750 x  - 5151296875 x + 12771880859375"
      ],
      "text/x-maxima": [
       "x^3+3491750*x^2-5151296875*x+12771880859375"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FindClassPolyD(x,D);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{15}$}\\frac{125\\,\\left(49\\,\\left(351747\\,\\sqrt{23}-371251163\\,\\sqrt{3}\\right)^{\\frac{2}{3}}-13967\\,2^{\\frac{4}{3}}\\,3^{\\frac{1}{6}}\\,\\left(351747\\,\\sqrt{23}-371251163\\,\\sqrt{3}\\right)^{\\frac{1}{3}}+15944845\\,2^{\\frac{2}{3}}\\,3^{\\frac{1}{3}}\\right)}{2^{\\frac{1}{3}}\\,3^{\\frac{7}{6}}\\,\\left(351747\\,\\sqrt{23}-371251163\\,\\sqrt{3}\\right)^{\\frac{1}{3}}}\\]"
      ],
      "text/plain": [
       "                                                     2/3\n",
       "(%o15) (125 (49 (351747 sqrt(23) - 371251163 sqrt(3))\n",
       "          4/3  1/6                                      1/3\n",
       " - 13967 2    3    (351747 sqrt(23) - 371251163 sqrt(3))\n",
       "             2/3  1/3     1/3  7/6                                      1/3\n",
       " + 15944845 2    3   ))/(2    3    (351747 sqrt(23) - 371251163 sqrt(3))   )"
      ],
      "text/x-maxima": [
       "(125*(49*(351747*sqrt(23)-371251163*sqrt(3))^(2/3)\n",
       "     -13967*2^(4/3)*3^(1/6)*(351747*sqrt(23)-371251163*sqrt(3))^(1/3)\n",
       "     +15944845*2^(2/3)*3^(1/3)))\n",
       " /(2^(1/3)*3^(7/6)*(351747*sqrt(23)-371251163*sqrt(3))^(1/3))"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exact_modular_j(p),factor;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{16}$}-3.4932256999699333682055047385473_B \\times 10^{6}\\]"
      ],
      "text/plain": [
       "(%o16)               - 3.4932256999699333682055047385473b6"
      ],
      "text/x-maxima": [
       "-3.4932256999699333682055047385473b6"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bfloat(%),fpprec:32;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{17}$}-3.4932256999699333682055047385475_B \\times 10^{6}\\]"
      ],
      "text/plain": [
       "(%o17)               - 3.4932256999699333682055047385475b6"
      ],
      "text/x-maxima": [
       "-3.4932256999699333682055047385475b6"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bfloat_modular_j(p),fpprec:32;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{18}$}0\\]"
      ],
      "text/plain": [
       "(%o18)                                 0"
      ],
      "text/x-maxima": [
       "0"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FindClassPolyD(x,D),x:bfloat_modular_j(p),fpprec:32;"
   ]
  }
 ],
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