{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Playing with imaginary quadratic irrationals\n",
    "Yasuaki Honda at gmail dot com\n",
    "\n",
    "- Two imaginary quadratic irrationals \\\\(\\alpha\\\\) and \\\\(\\beta\\\\) are equivalent iff there is a modular transformation f(z) in \\\\(SL_2(Z)\\\\).\n",
    "- Two imaginary quadratic irrationals are equivalent, their determinant are equal.\n",
    "- However, even if the determinant of two imaginary quadratic irrationals are equal, they may not be equivalent.\n",
    "- For a given negative determinant, there are finitely many imaginary quadratic irrationals whose determinant are equal to the given one. The number of such imaginary quadratic irrationals is called class number."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "install_github(\"YasuakiHonda\",\"modular_j\",\"main\")$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "asdf_load(\"modular_j\")$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{15}$}\\frac{\\sqrt{35}\\,i+1}{6}\\]"
      ],
      "text/plain": [
       "                                sqrt(35) %i + 1\n",
       "(%o15)                          ---------------\n",
       "                                       6"
      ],
      "text/x-maxima": [
       "(sqrt(35)*%i+1)/6"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a:(1+sqrt(-35))/6;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{16}$}\\frac{\\sqrt{35}\\,i-1}{6}\\]"
      ],
      "text/plain": [
       "                                sqrt(35) %i - 1\n",
       "(%o16)                          ---------------\n",
       "                                       6"
      ],
      "text/x-maxima": [
       "(sqrt(35)*%i-1)/6"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FindInFD(a);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{17}$}f\\left(x\\right):=\\frac{3\\,x+5}{1\\,x+2}\\]"
      ],
      "text/plain": [
       "                                        3 x + 5\n",
       "(%o17)                          f(x) := -------\n",
       "                                        1 x + 2"
      ],
      "text/x-maxima": [
       "f(x):=(3*x+5)/(1*x+2)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x):=(3*x+5)/(1*x+2);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{18}$}\\frac{\\sqrt{35}\\,i+89}{34}\\]"
      ],
      "text/plain": [
       "                               sqrt(35) %i + 89\n",
       "(%o18)                         ----------------\n",
       "                                      34"
      ],
      "text/x-maxima": [
       "(sqrt(35)*%i+89)/34"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(a),ratsimp,algebraic:true;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{19}$}\\mathbf{false}\\]"
      ],
      "text/plain": [
       "(%o19)                               false"
      ],
      "text/x-maxima": [
       "false"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "InFD(%);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Ta: \\(3\\)"
      ],
      "text/plain": [
       "inline-value"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "S"
      ],
      "text/plain": [
       "inline-value"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "Ta: \\(2\\)"
      ],
      "text/plain": [
       "inline-value"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{20}$}\\frac{\\sqrt{35}\\,i-1}{6}\\]"
      ],
      "text/plain": [
       "                                sqrt(35) %i - 1\n",
       "(%o20)                          ---------------\n",
       "                                       6"
      ],
      "text/x-maxima": [
       "(sqrt(35)*%i-1)/6"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FindInFD(f(a));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{21}$}-35\\]"
      ],
      "text/plain": [
       "(%o21)                               - 35"
      ],
      "text/x-maxima": [
       "-35"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FindDetIrr(a);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{22}$}\\left[ \\frac{\\sqrt{35}\\,i-1}{6} , \\frac{\\sqrt{35}\\,i-1}{2} \\right] \\]"
      ],
      "text/plain": [
       "                       sqrt(35) %i - 1  sqrt(35) %i - 1\n",
       "(%o22)                [---------------, ---------------]\n",
       "                              6                2"
      ],
      "text/x-maxima": [
       "[(sqrt(35)*%i-1)/6,(sqrt(35)*%i-1)/2]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FindReducedQuadIrrD(-35);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{23}$}\\left[ \\mathbf{true} , \\mathbf{true} \\right] \\]"
      ],
      "text/plain": [
       "(%o23)                           [true, true]"
      ],
      "text/x-maxima": [
       "[true,true]"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "map(InFD,%);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{3}$}2\\]"
      ],
      "text/plain": [
       "(%o3)                                  2"
      ],
      "text/x-maxima": [
       "2"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FindClassNumberD(-35);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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