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%$BF|K\8l$rB@;z$K$9$kL?Na$,(B \gt $B$G$"$k(B TeX $B$N>l9g$O(B, $B<!9T$N(B % $B$r=|$$$F$/$@$5$$(B
%\def\dg{\gt}

\def\ep{\varepsilon}
\def\F{\Bbb F}
\def\N{\Bbb N}
\def\Q{\Bbb Q}
\def\Z{\Bbb Z}
\def\gerC{\frak C}
\def\germ{\frak m}
\def\calO{{\cal O}}
\def\gerp{\frak p}
\def\gerP{\frak P}
\def\qed{\hfill $\Box$}
\def\cbrt#1{\sqrt[3]{#1}}
\def\GL{\operatorname{GL}}
\def\Gal{\operatorname{Gal}}
\def\rank{\operatorname{rank}}
\def\bysame{\leavevmode\hbox to 3em{\hrulefill}\,}

\topmargin-.3in
\oddsidemargin 0cm \evensidemargin 0cm
\textheight23.6cm
\textwidth16.2cm
\parindent1zw

\makeatletter
\def\@begintheorem#1#2{\trivlist \item[\hskip\labelsep\hskip\parindent{\bf #1\,#2.}]}
\def\@opargbegintheorem#1#2#3{\trivlist \item[\hskip \labelsep\hskip\parindent{\bf #1\,#2\,(#3).}]}
\makeatother
\newtheorem{th}{\dg $BDjM}(B}[section]
\newtheorem{lemma}[th]{\dg $BJdBj(B}
\newtheorem{prop}[th]{\dg $BL?Bj(B}
\newtheorem{cor}[th]{\dg $B7O(B}

\begin{document}

\title{$B<BFs<!BN>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~(B}
\author{$BAa0pEDBg3XM}9)3XIt(B\quad $B2C@n(B $B5.>O(B(Tak$@,$aaki KAGAWA)}
\date{}
\maketitle

\section{Introduction}

$k$$B$rBe?tBN$H$9$k(B.
$k$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~(B
($B$N(B$k$$B>e$NF17?N`(B)$B$rA4$F7hDj$9$k$3$H$rLdBj$H$9$k(B.

$B$3$l$^$G$K$O<!$N$h$&$J7k2L$,F@$i$l$F$$$k(B: \par
\medskip
(1) $\Q$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$OB8:_$7$J$$(B. (Tate$B$K$h$k(B.
$B:#$dH`$N(BWiles$B$NDjM}$N7O$G$"$k(B.) \par
(2) $k$$B$,N`?t$,(B6$B$HAG$J5uFs<!BN$J$i(B$k$$B>e$K$OB8:_$7$J$$(B
    (\cite{Setzer1}). \par
(3) $B?'!9$J<BFs<!BN>e$KNc$,$"$k(B
(\cite{Com}, \cite{Cremona_92}, \cite{Ishii}, \cite{Kida2},
 \cite{Pinch_thesis}, \cite{Setzer}, \cite{Shiota}, \cite{Ume}). \par
(4) $\Q(\sqrt 5)$ $B>e$K$OB8:_$7$J$$(B(\cite{Pinch_thesis}). \par
(5) $p$$B$,(B8$B$rK!$H$7$F(B5$B$H9gF1$JAG?t$N;~(B,
    $\Q(\sqrt p)$$B>eHsB8:_$+$I$&$+$N5,=`$,$"$k(B.
    $B$=$l$rMQ$$$F(B, $p=5,13$$B$N;~B8:_$7$J$$$3$H$,<($5$l$F$$$k(B
    (\cite{Ishii}). \par
(6) (5)$B$G=R$Y$?5,=`$,(B$\Q(\sqrt m)$, $m\equiv1\pmod4$$B$N>l9g$K3HD%$5$l$F$$$k(B.
    $B$=$l$rMQ$$$F(B$m=5,13,17,21,73,97,149,173,181$$B$N>l9g$NHsB8:_$N>ZL@$H(B,
    $m=41$$B>e$NA46J@~$N7hDj$,$J$5$l$F$$$k(B.
    (\cite{K-K})

\bigskip
$BK\9F$G$O<BFs<!BN$N>l9g$r07$&(B.
$B$3$N>l9g;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$H$7$F$O(B,
$BFs<!;XI8(B$\chi_N$ ($N>1$$B$O4pK\H=JL<0(B)
$B$r(BNeben-type character$B$K;}$DJ]7?7A<0$N6u4V(B$S_N:=S_2(\varGamma_0(N),\chi_N)$
$B$NFs<!85(B$\Q$-$BC1=cItJ,6u4V$+$iF@$i$l$k(B
Shimura's elliptic curve (cf.\,\cite{Shimura_bon})$B$b$"$j6=L#?<$$(B.
$B>ZL@$9$k$N$O<!$N$b$N$G$"$k(B:

\bigskip
(1)
$m=2$, 3, 5, 10, 13, 15, 17, 21, 30, 34, 39, 42, 47, 53, 58, 66, 69,
70, 73, 74, 85, 89, 94, 97$B$N;~(B,
$\Q(\sqrt m)$$B>e;j$k=j(Bgood reduction $B$r;}$DBJ1_6J@~$OB8:_$7$J$$(B. \par
(2)
$m=6$, 7, 14, 29, 33, 37, 41, 65$B$N;~(B,
$k=\Q(\sqrt m)$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$rA4$F7hDj$7$?(B
(Appendix 2$B$NDL$j(B).
$k$-isogeny class$B$N?t$O(B, $m=65$$B$N;~$@$1(B2$B$G(B, $BB>$N(B$m$$B$N;~$O(B1$B$G$"$k(B.

\bigskip
$B$3$N$&$A0lIt$O(B
\cite{K-K}$B$GF@$i$l$F$$$k$,(B, $B>ZL@$O$3$3$GM?$($i$l$k$b$N$NJ}$,0W$7$$(B.
$B$^$?(B$m=2,3,6,7,14,47,94$$B$N>l9g$OEE5$DL?.Bg3X$NLZED2m@.;a(B
(\cite{Kida_ram}, \cite{Kida_sympo})$B$K$h$jFHN)$KF@$i$l$F$$$k(B.

$B<BFs<!BN(B$\Q(\sqrt m)$ ($1<m<100$)$B$O(B60$B8D$"$k(B.
$B$3$3$G$O$=$N$&$A(B24$B8D$NBN$KBP$7HsB8:_$r<($7(B,
8$B8D$NBN$KBP$7$=$N>e$NBJ1_6J@~$rA4$F7hDj$7$F$$$k(B.
$B$3$3$G07$($J$+$C$?>l9g$GNc$,$"$k$N$O(B$m=22,26,38,77,79,86$$B$N>l9g$G$"$j(B,
$B;D$j$N(B22$B8D$N(B$m$$B$KBP$7(B,
$\Q(\sqrt m)$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$,B8:_$9$k$+$I$&$+$O(B
$B$^$@$o$+$i$J$$(B($BB?J,L5$$(B).

Shimura's ellptic curve$B$H$N4X78$r5s$2$F$*$3$&(B.
$d(m)$$B$rFs<!BN(B$\Q(\sqrt m)$$B$NH=JL<0$H$9$k(B.
$B$3$N;~(B$S_{d(m)}$$B$N9=B$$,CN$i$l$F$$$k(B.
$m$$B$,(B(1)$B$K5s$2$i$l$F$$$k$$$:$l$+$NCM$N;~(B,
$S_{d(m)}$$B$O(B2$B<!85$N(B$\Q$-$BC1=cItJ,6u4V$r;}$?$J$$(B.
$m=6,7,29,33,37,41$$B$KBP$7$F$O(B,
$S_{d(m)}$$B$O(B2$B<!85(B$\Q$-$BC1=c$G$"$j(B,
$S_{d(14)}$$B$O(B2$B<!85(B$\Q$-$BC1=cItJ,6u4V$H(B4$B<!85(B$\Q$-$BC1=cItJ,6u4V$ND>OB(B,
$S_{d(65)}$$B$O(B2$B$D$N(B2$B<!85(B$\Q$-$BC1=cItJ,6u4V$ND>OB$G$"$k(B.
($B>e$N(B$S_{d(m)}$$B$N7W;;$OD9C+@nM:G7;a$HF|HfLn9d;N;a$K$h$k(B.)
$B$h$C$F>e$N(B32$B8D$N(B$m$$B$KBP$7(B,
$BA4$F$N6J@~$O(BShimura's ellpitic curve, $BFC$K(Bmodular$B$G$"$k(B.
($m=29,37,41,65$$B$N>l9g$O(B, $B$I$N6J@~$b5u?t>hK!$r;}$C$F$$$J$$$3$H$KCm0U(B.)

\bigskip
{\dg $B5-9f(B}:
$BBe?tBN(B$k$$B$KBP$7(B,
$\calO_k$, $\calO_k^\times$, $h_k$$B$G$=$l$>$l(B
$k$$B$N@0?t4D(B, $BC1?t72(B, $BN`?t$rI=$9$H$9$k(B.
$\germ$$B$,(B$k$$B$N0x;R(B($BB($AJ,?t(Bideal$B$H$$$/$D$+$NL58BAGE@$N7A<0E*$J@Q(B)$B$N;~(B,
$h_k(\germ)$$B$r(B$k$$B$N(Bray class number modulo $\germ$$B$H$9$k(B.
$B4JC1$N$?$a(B$h_k(\prod_{\gerp|2}\gerp)$$B$r(B$h_k^{(2)}$$B$H=q$/(B.
$B$^$?(B$k$$B$,<BFs<!BN$J$i$P(B, $k/\Q$$B$N6&Lr$r(B$'$$B$GI=$7(B,
$\ep$ ($>1$)$B$G4pK\C1?t$rI=$9$H$9$k(B.
$k=\Q(\sqrt m)$, $m\equiv1\pmod4$$B$N;~$O(B, $\omega=(1+\sqrt m)/2$$B$H$9$k(B.

$BBJ1_6J@~(B$E$$B$KBP$7$F(B,
$b_2(E)$, $b_4(E)$, $b_6(E)$, $b_8(E)$, $c_4(E)$, $c_6(E)$, $\varDelta(E)$,
$j(E)$$B$rDL>oDL$j$H$9$k(B.
($BDj5A$O(B\cite{Sil1}$B$r;2>H(B.
 $j$$B0J30$O%b%G%k$K0MB8$9$k$N$G(B,
 $b_2(E)$$BEy$O0l$D%b%G%k$r7h$a$F9M$($?$b$N$HN;2r$7$F$$$?$@$-$?$$(B).
$n\in\N$$B$KBP$7(B$E[n]$$B$r(B$nP=O$$B$J$kE@(B$P$$BA4BN$N@.$972$H$9$k(B.
$E$$B$,(B$k$$B>eDj5A$5$l$F$$$k;~(B,
$k(E[n])$$B$r(B$E[n]$$B$N(B$O$$B$G$J$$A4$F$NE@$N:BI8$r(B$k$$B$KE:2C$7$?BN$H$9$k(B.

\section{Some criteria}
$B$3$N@a$G$O4v$D$+5,=`$r>R2p$9$k(B.

\begin{prop} \label{prop1}
$k$$B$rBe?tBN(B, $E$$B$r(B$k$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$H$9$k(B.
$E$$B$,0L?t(B2$B$N(B$k$-$BM-M}E@$r;}$?$J$1$l$P(B,
$h_{k(\sqrt{\varDelta\smash{(E)}})}^{(2)}$$B$O(B3$B$G3d$l$k(B. \qed
\end{prop}

\begin{cor} \label{criterion2}
$k$$B$r(B$(h_k,6)=1$, $B$^$?$O(B$h_k=2$$B$J$k<BFs<!BN$H$9$k(B.
$h_k^{(2)}$, $h_{k(\sqrt{-1})}^{(2)}$, $h_{k(\sqrt{\pm\ep})}^{(2)}$
$B$,A4$F(B3$B$G3d$l$J$1$l$P(B,
$k$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$OA4$F(Badmissible$B$G$"$k(B. \qed
\end{cor}

$B$3$3$G(B, $BBe?tBN(B$k$$B>eDj5A$5$l$?BJ1_6J@~(B$E$$B$,(Badmissible$B$G$"$k$H$O(B,
$E$$B$,<!$rK~$?$9$3$H$r8@$&(B:\par
(1) $E$$B$O(B$k$$B>e;j$k=j(Bgood reduction$B$r;}$D(B; \par
(2) $E$$B$O0L?t(B2$B$N(B$k$-$BM-M}E@$r;}$D(B.

\begin{prop} \label{criterion3}
$k$$B$r<BFs<!BN$H$9$k(B.
$h_k((3)\gerp_\infty^{(1)}\gerp_\infty^{(2)})$
($\gerp_\infty^{(1)}$, $\gerp_\infty^{(2)}$$B$O(B$k$$B$NL58BAGE@(B)
$B$,(B4$B$G3d$l$J$1$l$P(B,
$k$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$GH=JL<0$,(B$k$$B$N(B3$B>h?t$N$b$N$O(B
$k$$B>eDj5A$5$l$?(B$3$-isogeny$B$r5v$9(B.
\qed
\end{prop}

\begin{prop} \label{criterion3'}
$k$$B$r<BFs<!BN$H$9$k(B.
12$B<!BN(B$k(\cbrt{\ep},\sqrt{-3})$$B$N(B
3$B$rK!$H$9$k(Bray class number$B$,4q?t$G$"$k$H2>Dj$9$k(B.
$k$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~(B$E$$B$NH=JL<0(B$\varDelta(E)$$B$,(B
$k$$B$N(B3$B>h?t$G$J$$$J$i$P(B,
$\Gal(k(E[3])/k)$ ($\subset\GL_2(\F_3)$)$B$O(B
\[ \begin{pmatrix}
     1 & * \\ 0 & *
   \end{pmatrix}, \quad
   \begin{pmatrix}
     * & * \\ 0 & 1
   \end{pmatrix}
\]
$B$N$$$:$l$+$H6&Lr$G$"$k(B.
$BFC$K(B, $E$$B$O0L?t(B3$B$N(B$k$-rational subgroup $V$$B$r;}$A(B,
$V\subset E(k)$$B$G$"$k(B($BB($A(B$E$$B$,0L?t(B3$B$N(B$k$-$BM-M}E@$r;}$D(B)$B$+(B,
$E/V$$B$,0L?t(B3$B$N(B$k$-$BM-M}E@$r;}$D$+$N$I$A$i$+$G$"$k(B. \qed
\end{prop}

\begin{prop}  \label{criterion4}
$B<BFs<!BN(B$k$$B$,<!$rK~$?$9;~(B,
$k$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$NH=JL<0$O(B$k$$B$N(B3$B>h?t$G$"$k(B: \par
(1) $(h_k,6)=1$; \par
(2) $k$$B$K$*$$$F(B$3$$B$OITJ,4t$G$"$k(B; \par
(3) $h_{k(\sqrt{-3})}$$B$O(B3$B$G3d$l$J$$(B; \par
(4) $h_{k(\cbrt{\ep})}$$B$NN`?t$O(B2$B$G3d$l$J$$(B; \par
(5) $3$$B$r3d$k(B$k$$B$NAG(Bideal $\gerp$$B$KBP$7(B,
    $B9gF1<0(B$X^3\equiv\ep \pmod{\gerp^2}$$B$O2r(B$X\in\calO_k$$B$r;}$?$J$$(B. \qed
\end{prop}

$B>ZL@$NJ}?K$O$$$:$l$bF1$8$G$"$k(B. $BB($A(B
\begin{itemize}
\item $BBe?tBN(B$k$$B>eDj5A$5$l$?BJ1_6J@~$H<+A3?t(B$n$$B$KBP$7(B,
      $\Gal(k(E[n])/k)$$B$O(B$\GL_2(\Z/n\Z)$$B$NItJ,72$KF17?$G$"$k(B;
\item $E$$B$,Be?tBN(B$k$$B>e;j$k=j(Bgood reduction$B$r;}$D$H$-(B,
      $k(E[n])/k$$B$GJ,4t$9$k2DG=@-$N$"$kM-8BAGE@$O(B
      $n$$B$NAG0x;R$H(B$E$$B$N(Bbad prime$B$NAG0x;R$N$_$G$"$k(B,
\item $k(E[n])$ ($n=2,3$)$B$O(B$\sqrt[n]{\varDelta\smash{(E)}}$,
      1$B$N86;O(B$n$$B>h:,$r4^$`(B
\end{itemize}
$B$H$$$&$h$/CN$i$l$?;v<B$r;H$C$F(B,
$n$$B$HL58BAGE@$N30ITJ,4t$J(Babel$B3HBg(B, $B$"$k$$$OITJ,4t(Babel$B3HBg$r:n$k(B,
$B$=$N$h$&$JBN$NB8:_$ON`BNO@$K$h$j(B(ray) class number$B$N2D=|@-$K5"Ce$5$l$k(B,
$B$H$$$&$o$1$G$"$k(B.
$B>\:Y$O(B\cite{ore}, \cite{ore_29}, \cite{ore_33}$B$r8fMw$$$?$@$-$?$$(B
(\cite{K-K}, \cite{Kida_ram}, \cite{Kida_sympo}$B$b(B).

$B$J$*$$$/$D$+$N5,=`$N>r7o$K$"$k(B``$(h_k,6)=1$''$B$O(B,
global minimal model$B$NB8:_$N$?$a$K;H$o$l$F$$$k(B. $BB($A(B

\begin{lemma}[Setzer \cite{Setzer1}] \label{lemma:global_minimal}
$k$$B$rBe?tBN(B, $E$$B$r(B$k$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$H$9$k(B.
$(h_k,6)=1$$B$J$i$P(B$E$$B$O(Bglobal minimal model$B$r;}$D(B. \qed
\end{lemma}

\section{Determination}

$BA0@a$N5,=`$r;H$C$F(B, $B<BFs<!BN>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$r7hDj$9$k(B.
$B5,=`$N2>Dj$r3NG'$9$k$K$O(Bray class number$B$N7W;;$r$7$J$/$F$O$$$1$J$$$,(B,
$B$=$N7W;;$O(BKASH$B$r;H$($P=PMh$k(B.
(KASH$B$K$D$$$F$O(B, $B@):n<T$?$A$N=q$$$?>\$7$$2r@b(B\cite{KANT}$B$,$"$k(B.)

\subsection{Admissible curves} \label{only2}

$k=\Q(\sqrt{m})$ ($1<m<100$)$B$rN`?t$,(B6$B$HAG(B,
$B$^$?$ON`?t$,(B2$B$G$"$k<BFs<!BN$H$9$k(B
($k$$B$NN`?t$,(B3$B$J$N$O(B$m=79$$B$N;~$N$_(B, $BN`?t$,(B4$B$J$N$O(B$m=82$$B$N;~$N$_$G(B,
 $BB>$N;~$ON`?t(B1$B$+(B2$B$G$"$k(B).
$h_K^{(2)}$ ($K=k$, $k(\sqrt{-1})$, $k(\sqrt{\pm\ep})$)$B$r(B
$B7W;;$9$k$HI=(B\ref{h_K(2)}\,$B$N$h$&$K$J$k(B(3$B$NG\?t$OB@;z$K$7$?(B).

{\small
\makeatletter
\def\@captype{table}
\makeatother
\begin{center}
\begin{tabular}{|r|c|c|c|c||r|c|c|c|c|}
\hline
$m$ & $k$ & $k(\sqrt{-1})$ & $k(\sqrt \ep)$ &  $k(\sqrt{-\ep})$ &%
$m$ & $k$ & $k(\sqrt{-1})$ & $k(\sqrt \ep)$ &  $k(\sqrt{-\ep})$ \\
\hline
 2 & 1 & 1  & 1  &  1  & 47 & 1  &   5  &  1  &  8  \\
 3 & 1 & 1  & 1  &  2  & 51 & 2  &  {\bf 12} & 4 & 8 \\
 5 & 1  &   1  &  1  &  1  & 53 & 1  &  {\bf 3} & 1 & 1 \\
 6 & 1  &   2  &  1  &  1  & 55 &2 & 4 & 1 & {\bf 12} \\
 7 & 1  &   1  &  1  &  4  & 57 &1 &  2 & 1 & {\bf 3} \\
10 & 2  &   2  &  2  &  2  & 58 &2 &  2 & 2 & 2 \\
11 & 1  &   {\bf 3}  &  1  &  2 & 59 &1 & {\bf 9} & 1 & {\bf 6} \\
13 & 1  &   1  &  1  &  1 & 61 &1 & {\bf 3} & 1 & 1 \\
14 & 1  &   4  &  1  &  1  & 62 &1  &  8  &  1  & {\bf 3} \\
15 & 2  &   2  &  2  &  8  & 65 & 2  & 8 &  2  &  2 \\
17 & 1  &   2  &  1  &  1  & 66 & 2  & 16 &  1  & 8 \\
19 & 1  & {\bf  3}  &  1  & {\bf  6} & 67 & 1  & {\bf 3} &  1 &  14 \\
21 & 1  &   2  &  1  &  1  & 69      &1 &  4 & 1 & {\bf 3} \\
22 & 1  &   2  &  1  & {\bf 3} & 70  &2 &  4 & 1 & 16 \\
23 & 1  & {\bf 3} &  1  &  4  & 71      &1 & 7 & {\bf 3} & 4 \\
26 & 2  & {\bf 6} &  2  &  2  & 73      &1 &  2 & 1 & 1 \\
29 & 1  & {\bf 3} &  1  &  1  & 74      &2 & 10 & 2 & 2 \\
30 & 2  & 4 &  1  & 8 & 77      &1 &  4 & 1 & {\bf 3} \\
31 & 1  & {\bf 3} &  1  &  8 & 78      &2 &  4 & 2 & {\bf 12} \\
33 & 1  &   2  &  1  & {\bf 3} &83      &1 & {\bf 9} & 1 & 10 \\
34 & 2  &  8 &  1  &  8 & 85  &2 &  4 & 4 & 4 \\
35 & 2  &  {\bf 6} & 2 & 16 & 86      &1 & 10 & 1 & {\bf 3} \\
37 & {\bf 3} & {\bf 3} & {\bf 3} & {\bf 3} & 87 &2 & {\bf 6} & 2 & 8 \\
38 & 1  & {\bf 6} &  1  & {\bf 3} & 89      &1 & {\bf 6} & 1 & 1 \\
39 & 2  &   4  &  1  & 4 & 91      &2 &  {\bf 6} & 2 & {\bf 48} \\
41 & 1  &   4  &  1  &  1  & 93      &1 &  2 & 1 & {\bf 3} \\
42 & 2  &  4 &  2  & 4 & 94      &1 & 8 & 1 & 5 \\
43 & 1  & {\bf  3} &  1  &  10  & 95      &2 &  8 & 1 & {\bf 12} \\
46 & 1  &   4  &  1  & {\bf 3} &  97      &1 &  2 & 1 & 1 \\
\hline
\end{tabular}
\end{center}
\caption{$h_K^{(2)}$ ($K=k,k(\protect\sqrt{-1}),k(\protect\sqrt{\pm\ep})$)}
\label{h_K(2)}
}

\bigskip
$B$3$N7W;;$H7O(B\ref{criterion2}\,$B$h$j<!$,F@$i$l$k(B:

\begin{prop} \label{admitting_adm}
$m$$B$,(B
 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 21, 30, 34, 39, 41, 42, 47, 58, 65, 66,
70, 73, 74, 85, 94, 97
$B$N$$$:$l$+$J$i$P(B,
$\Q(\sqrt m)$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$OA4$F(Badmissible$B$G$"$k(B. \qed
\end{prop}

Comalada \cite{Com}$B$O(B,
$B$=$N>e$K(Badmissible curve$B$r5v$9$h$&$J<BFs<!BN$NFCD'IU$1$rITDjJ}Dx<0$rMQ$$$FM?$($F$$$k(B. $B$=$7$FITDjJ}Dx<0$r<B:]$K2r$/$3$H$K$h$j(B,
$k=\Q(\sqrt m)$ ($1<m<100$)$B>e(Badmissible$B$JBJ1_6J@~$,B8:_$9$k$N$O(B
\[ m=6,7,14,22,38,41,65,77,86 \]
$B$N>l9g$K8B$k$3$H$r<($7$F$$$k(B.
$B99$K$3$N;~(B
admissible$B$JBJ1_6J@~$N(B$k$-i\-so\-mor\-phism class$B$,A4$F5a$a$i$l$F$$$k(B.
$B$=$l$HL?Bj(B\ref{admitting_adm}\,$B$r$"$o$;$F<!$,F@$i$l$k(B:

\begin{cor} \label{nonexistence:1}
(1)
$m=2$, 3, 5, 10, 13, 15, 17, 21, 30, 34, 39, 42, 47, 58, 66,
70, 73, 74, 85, 94, 97$B$N;~(B,
$\Q(\sqrt m)$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$OB8:_$7$J$$(B. \par
(2) $m=6$, 7, 14, 41, 65$B$N;~(B,
$k=\Q(\sqrt m)$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$OA4$F(Badmissible$B$G$"$k(B.
$B=>$C$F(BAppendix 2$B$NI=$K$"$k$b$N$@$1$G$"$k(B.
(Comalada$B$NBJ1_6J@~$H$NBP1~$O(BAppendix 2$B$NI=$K=q$$$F$*$$$?(B.)
\qed
\end{cor}

Comalada$B$N7k2L(B, $BI=(B\ref{h_K(2)},
$BL?Bj(B\ref{prop1}, $BJdBj(B\ref{lemma:global_minimal}\,$B$r$"$o$;$F<!$bF@$i$l$k(B:

\begin{cor}
$E$$B$r<BFs<!BN(B$k=\Q(\sqrt m)$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$H$9$k(B. \par
(1) $m$$B$,(B33$B$^$?$O(B69$B$N;~(B,
$E$$B$O(B$\varDelta(E)=-\ep^{2n+1}$ ($n\in\Z$)$B$J$k(Bglobal minimal model$B$r;}$D(B. \par
(2) $m$$B$,(B$29,53,89$$B$N$$$:$l$+$G$"$k;~(B,
$E$$B$O(B$\varDelta(E)=-\ep^{2n}$ ($n\in\Z$)$B$J$k(Bglobal minimal model$B$r;}$D(B. \qed
\end{cor}

$\Q(\sqrt{37})$$B>e$N6J@~$KBP$7$F$O(B,
$BF1$8J}K!$G$OH=JL<0$NId9f$b;X?t$N6v4q$b$o$+$i$J$$(B.
$B$H$$$&$N$O(B, $h_k^{(2)}$, $h_{k(\sqrt{-1})}^{(2)}$, $h_{k(\sqrt{\pm\ep})}^{(2)}$
$B$,A4$F(B3$B$@$+$i$G$"$k(B.

\section{$m=29,33,69$$B$N>l9g(B}

$B0J2<(B$k=\Q(\sqrt m)$, $1<m<100$, $(h_k,6)=1$$B$H$9$k(B.
$B$^$?(B$m$$B$O:#$^$G07$o$l$F$$$J$$$b$N$H$9$k(B.
$BB($A(B
$m=11$, 19, 22, 23, 29, 31, 33, 37, 38, 43, 46, 53, 57, 59, 61, 62, 67, 69,
71, 77, 83, 86, 89, 93.
$B$3$N;~(B$h_k=1$$B$G$"$k(B.

$B$^$:H=JL<0$,(B3$B>h?t$N>l9g$r07$&(B.
$B:G=i$K<!$NJdBj$r5s$2$F$*$/(B($B>ZL@$O(B\cite{ore_33}$B$r;2>H$N$3$H(B):

\begin{lemma} \label{prop2}
$k$$B$rN`?t$,(B6$B$HAG$J<BFs<!BN(B,
$E$$B$r(B$k$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$H$9$k(B.
$B$3$N;~(B$E$$B$,(B$k$$B>eDj5A$5$l$?(B3-isogeny$B$r5v$;$P(B,
$k$$B$O(B$\Q(\sqrt 6)$$B$+(B$\Q(\sqrt{33})$$B$N$$$:$l$+$G(B,
$E$$B$O(B6A1, 6A1$'$, $BKt$O(B33A1, 33A1$'$$B$H$=$l$>$l$NBN>eF17?$G$"$k(B. \qed
\end{lemma}

$BL?Bj(B\ref{criterion3}\,$B$r;H$&$?$a$K(B
$h_k((3)\gerp_{\infty}^{(1)}\gerp_{\infty}^{(2)})$$B$r5a$a$k(B.
$B$=$NA0$K(B, $B<!$rCm0U$7$F$*$/(B.
$p$$B$rAG?t$H$9$k;~(B,
$k$$B$,(B$\Q(\sqrt p)$ ($p\equiv1\pmod4$),
$\Q(\sqrt{3p})$ ($p\equiv3\pmod4$)$B$N$$$:$l$G$b$J$$<BFs<!BN$N;~$O(B,
$h_k((3)\gerp_{\infty}^{(1)}\gerp_{\infty}^{(2)})$
$B$O(B4$B$NG\?t$G$"$k(B.
$B<B:]$3$N;~(B$\tilde k$$B$r(B$k$$B$N(Bnarrow genus field$B$H$9$k$H(B,
$\tilde k(\sqrt{-3})/k$$B$O(B3$B$HL58BAGE@$N30ITJ,4t$J(B4$B<!0J>e$N(Babel$B3HBg$G$"$k(B.
$BB>$N>l9g$r(BKASH$B$G5a$a$k$H(B
\[ h_k((3)\gerp_{\infty}^{(1)}\gerp_{\infty}^{(2)})=
   \begin{cases}
   2 & \text{($m=29,33,53,89$$B$N;~(B),} \\
   6 & \text{($m=69$$B$N;~(B),} \\
   12 & \text{($m=93$$B$N;~(B),} \\
   4 & \text{($B$=$NB>(B)}
   \end{cases}
\]
$B$,F@$i$l$k(B.
$B$h$C$FL?Bj(B\ref{criterion3}, $BJdBj(B\ref{prop2}\,$B$h$j<!$,=>$&(B.

\begin{prop} \label{prop11}
(1) $m=29,53,69,89$$B$N;~(B, $k=\Q(\sqrt m)$$B>e;j$k=j(Bgood reduction$B$r;}$A(B,
$BH=JL<0$,(B$k$$B$N(B3$B>h?t$G$"$kBJ1_6J@~$OB8:_$7$J$$(B. \par
(2) $k=\Q(\sqrt{33})$$B>e;j$k=j(Bgood reduction$B$r;}$A(B,
$BH=JL<0$,(B$k$$B$N(B3$B>h?t$G$"$kBJ1_6J@~$O(B33A1, 33A1$'$$B$N$_$G$"$k(B. \qed 
\end{prop}

$B<!$KH=JL<0$,(B3$B>h?t$G$J$$>l9g$r9M$($k(B.
$BL?Bj(B\ref{criterion3'}\,$B$r;H$&$N$G(B,
12$B<!BN(B$M=k(\cbrt{\ep},\sqrt{-3})$$B$N(B$(3)$$B$rK!$H$9$k(Bray class number $h_M((3))$$B$r(B
KASH$B$G5a$a$k$H0J2<$N$h$&$K$J$k(B:
\[ h_M((3))=
   \begin{cases}
        3   & (\text{$m=29$$B$N;~(B}), \\
        243 & (\text{$m=33$$B$N;~(B}), \\
        9   & (\text{$m=69$$B$N;~(B}), \\
        12  & (\text{$m=77$$B$N;~(B}), \\
        18  & (\text{$m=93$$B$N;~(B}). \\
   \end{cases}
\]
$B7W;;;~4V$O(BSparc station SS4 (CPU 110MHZ)$B$G$=$l$>$l(B
1$BJ,(B, 20$BJ,(B, 1$BJ,(B, 8$BJ,(B, 2$BJ,DxEY$G$"$C$?(B.
($BB>$N(B$m$$B$N>l9g$O(B, Minkowski bound$B$,Bg$-$/(B,
 $B;~4V(B, $B%a%b%j!<6&$K?)$$$=$&$J$N$G7W;;$7$J$+$C$?(B.)
$B$h$C$FL?Bj(B\ref{criterion3'}\,$B$h$j(B

\begin{prop} \label{prop12}
$k=\Q(\sqrt m)$ ($m=29,33,69$)$B$H$9$k(B.
$BBJ1_6J@~(B$E$$B$,(B$k$$B>e;j$k=j(Bgood reduction$B$r;}$A(B,
$B$=$NH=JL<0$,(B$k$$B$N(B3$B>h?t$G$J$1$l$P(B,
$E$$B$O(B$k$$B>eDj5A$5$l$?(B3-isogeny $f:E\to\bar E$$B$r5v$7(B,
$E$$B$+(B$\bar E$$B$N$$$:$l$+$,0L?t(B3$B$N(B$k$-$BM-M}E@$r;}$D(B. \qed
\end{prop}

$B=>$C$FL?Bj(B\ref{prop11}, \ref{prop12}\,$B$r9MN8$9$k$H(B,
$k=\Q(\sqrt m)$ ($m=29,33,69$)$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$r7h$a$k$K$O(B,$B0L?t(B3$B$N(B$k$-$BM-M}E@$r;}$DBJ1_6J@~$GH=JL<0$,(B$k$$B$N(B3$B>h?t$G$J$$$b$N$r7h$a$l$P$h$$(B.
$m=29$$B$N;~$O(B,
$B$=$N$h$&$J6J@~$O(B29A1, 29A1$'$$B$NFs$D$@$1$G$"$k$3$H$,(B
\cite{Nakamura}$B$G>ZL@$5$l$F$$$k(B.
$m=33,69$$B$N;~$O$J$$$3$H$,$o$+$k(B
(cf.\,\cite{ore_33}).
$B$h$C$F(B

\begin{th}
(1) $\Q(\sqrt{29})$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$O(B
29A1, 29A1$'$, 29A2, 29A2$'$$B$N(B4$BK\$@$1$G$"$k(B. \par
(2) $\Q(\sqrt{33})$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$O(B
33A1, 33A1$'$, 33A2, 33A2$'$, 33A3, 33A3$'$$B$N(B6$BK\$@$1$G$"$k(B. \par
(3) $\Q(\sqrt{69})$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$OB8:_$7$J$$(B. \qed
\end{th}

$\Q(\sqrt{29})$$B$N>l9g$O(B\cite{ore_29}$B$b8+$F$$$?$@$1$k$H9,$$$G$"$k(B.

\section{$m=53,89$$B$N>l9g(B}
$m$$B$rA0@a$N:G=i$K5s$2$?(B$m$$B$N$&$A(B$m\ne29,33,69$$B$J$k$b$N$H$9$k(B.
$B$3$N;~L?Bj(B\ref{criterion4}\,$B$N2>Dj$r3N$+$a$k(B.

$B4{=R$N$h$&$K(B,
$k=\Q(\sqrt m)$, $1<m<100$, $(h_k,6)=1$$B$J$i(B$h_k=1$$B$G$"$k(B.
$B$h$C$F(B(1), (2)$B$O(B$57,93$$B0J30$N(B$m$$B$KBP$7$F$O@.N)$7$F$$$k(B.
$k(\sqrt{-3})$, $k(\cbrt{\ep})$$B$NN`?t$r7W;;$9$l$P(B
$BI=(B\ref{Table2}\,$B$N$h$&$K$J$k(B($k(\cbrt{\ep})$$B$NN`?t$O?tIC$G5a$^$k(B).

{\small
\makeatletter
\def\@captype{table}
\makeatother
\begin{center}
\begin{tabular}{|c|c|c||c|c|c|}
\hline
$m$ & $k(\sqrt{-3})$ & $k(\cbrt{\ep})$ &%
$m$ & $k(\sqrt{-3})$ & $k(\cbrt{\ep})$ \\
\hline
11  &      2  & 1 & 59  &       2  &      1  \\
19  &      2  & 1 & 61  &       4  &      1  \\
22  &      4  & 1 & 62  & {\bf  6} &      1  \\
23  &      4  & 1 & 67  & {\bf  6} &      1  \\
31  &      2  & 1 & 71  &       4  & {\bf 2} \\
37  &      4  & 1 & 77  & {\bf  6} &      1  \\
38  &      4  & 1 & 83  & {\bf  6} &      1  \\
43  & {\bf 6} & 1 & 86  &       4  &      1  \\
46  &      4  & 1 & 89  &       1  &      1  \\
\cline{4-6}
53  &      5  & 1 & \multicolumn{3}{c}{ } \\
\cline{1-3}
\end{tabular}
\end{center}
\caption{$BN`?t(B} \label{Table2}
}

\bigskip
{\dg $BCm(B}.
$B>r7o(B(3)$B$O(B$\Q(\sqrt{-3m})$$B$NN`?t$,(B3$B$G3d$l$J$$$3$H$HF1CM$G$"$j(B
(cf.$\;$\cite{Kubota}, \cite{Masley}),
$B$3$N$3$H$r;H$($P7W;;NL$r8:$i$9$3$H$,=PMh$k(B.

\bigskip
$B$h$C$F(B(3), (4)$B$rK~$?$9(B$m$$B$O(B
\[ m=11,19,22,23,31,37,38,46,53,59,61,86,89 \]
$B$G$"$k(B.
(5)$B$bMF0W$K3NG'$G$-$k(B:
KASH$B$OAG?t$NAG(Bideal$BJ,2r(B, ideal$B$,C19`$+$I$&$+(B,
$B?t$,(Bideal$B$K4^$^$l$k$+$I$&$+Ey$r3NG'$G$-$k$N$G(B,
$B$A$g$C$H$7$?%W%m%0%i%`$r=q$1$P$h$$(B.
$B$7$+$7(B(5)$B$N3NG'$/$i$$$O<j$G$bBg$7$?$3$H$J$$$N$G(B, $B$d$C$F$7$^$*$&(B
($BI=(B\ref{Table3}, \ref{Table4}).
$3=\gerp\gerp'$ ($\gerp\ne\gerp'$)$B$N;~$O(B,
$(\calO_k/\gerp^2)^\times\cong(\Z/9\Z)^\times$$B$J$N$G(B,
$B$3$N>l9g(B(5)$B$O(B$\ep\not\equiv\pm1\pmod{\gerp^2}$$B$HF1CM$G$"$k$3$H$KCm0U(B.

\bigskip
{\small
\makeatletter
\def\@captype{table}
\makeatother
\newlength{\alength}
\settowidth{\alength}{$(11+2\sqrt{31})$}
\newlength{\blength}
\settowidth{\blength}{$\ep\bmod\gerp^2$}
\begin{center}
\begin{tabular}{|r|p{\alength}|l|p{\blength}|}
\hline
$m$ & \hfil$\gerp$\hfill & \hfil $\ep$ \hfill    & $\ep \bmod \gerp^2$ \\
\hline
 19 & $(4+\sqrt{19})$    & $170+39\sqrt{19}$     & \hfil$-4$ \\
 22 & $(5+\sqrt{22})$    & $197+42\sqrt{22}$     & \hfil$-4$ \\
 31 & $(11+2\sqrt{31})$  & $1520+273\sqrt{31}$   & \hfil$-4$ \\
 37 & $(3+\omega)$       & $5+2\omega$           & \hfil$-4$ \\
 46 & $(7+\sqrt{46})$    & $24335+3588\sqrt{46}$ & \hfil$\phantom{-}2$ \\
 61 & $(3+\omega)$       & $17+5\omega$          & \hfil$-4$ \\
\hline
\end{tabular}
\caption{$\ep\bmod\gerp^2$ ($3=\gerp\gerp')$} \label{Table3}
\end{center}
}

\bigskip
{\small
\makeatletter
\def\@captype{table}
\makeatother
\begin{center}
\begin{tabular}{|r|l|l|l|}
\hline
$m$ & \hfil$(\calO_k/9\calO_k)^{\times3}$\hfill & \hfil$\ep$\hfill & $\hfil\ep\bmod9$\hfill \\
\hline
 11 & $\pm1$, $\pm2\sqrt{11}$, $\pm(2\pm4\sqrt{11})$ & $10+3\sqrt{11}$ & $1+3\sqrt{11}$ \\
 23 & $\pm1$, $\pm4\sqrt{23}$, $\pm(2\pm\sqrt{23})$ & $24+5\sqrt{23}$ & $-3-4\sqrt{23}$ \\
 38 & $\pm1$, $\pm2\sqrt{38}$, $\pm(2\pm4\sqrt{38})$ & $37+6\sqrt{38}$ & $1-3\sqrt{38}$ \\
 53 & $\pm1$, $\pm4\omega$, $\pm(4-4\omega)$, $\pm(1-2\omega)$ & $3+\omega$ & $3+\omega$ \\
 59 & $\pm1$, $\pm4\sqrt{59}$, $\pm(2\pm\sqrt{59})$ & $530+69\sqrt{59}$ & $-1-3\sqrt{59}$ \\
 86 & $\pm1$, $\pm4\sqrt{86}$, $\pm(2\pm\sqrt{86})$ & $10405+1122\sqrt{86}$ & $1-3\sqrt{86}$ \\
 89 & $\pm1$, $\pm(4-4\omega)$, $\pm(1-2\omega)$, $\pm4\omega$ & $447+106\omega$ & $-3-2\omega$ \\
\hline
\end{tabular}
\caption{$\ep\bmod9$ (3$B$,(B$k$$B$GBF@-$7$F$$$k;~(B)} \label{Table4}
\end{center}
}

\bigskip
$B7k6I>e$N(B13$B8D$N(B$m$$B$O2>Dj(B(1)$B$+$i(B(5)$B$rA4$FK~$?$9$N$G(B, $B<!$,>ZL@$5$l$?(B:

\begin{prop} \label{all_are_cube}
$m=11$, 19, 22, 23, 31, 37, 38, 46, 53, 59, 61, 86, 89$B$N;~(B,
$k=\Q(\sqrt m)$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$NH=JL<0$O(B
$k$$B$N(B3$B>h?t$G$"$k(B. \qed
\end{prop}

$m=53,89$$B$N>l9g$OMF0W$KJRIU$/(B.

\begin{th}
$\Q(\sqrt m)$ ($m=53,89$)$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$OB8:_$7$J$$(B.
\end{th}

{\dg $B>ZL@(B}.
$BL?Bj(B\ref{prop11}, \ref{all_are_cube}\,$B$h$jL@$i$+$G$"$k(B.
\qed

\section{$m=37$$B$N>l9g(B}

$k=\Q(\sqrt{37})$$B$N>l9g$O(B\cite{Kida2}$B$K$*$$$F(B,
$k$$B>e;j$k=j(Bgood reduction$B$r;}$A(B,
$B$7$+$b(B$j$-invariant$B$,M-M}@0?t$G$"$k6J@~$,7hDj$5$l$F$$$k(B.
$B>r7o(B``$j\in\Z$''$B$r=|$-$?$$$N$G(B,
$\Q(\sqrt{11})$$B$J$I$h$j$b$"$($F(B$\Q(\sqrt{37})$$B$rM%@h$7$?(B.
$BB>$NBN$KBP$7$F$O(B, $B$^$?JL=j$GH/I=$9$k$D$b$j$G$"$k(B.

$E$$B$r(B$k$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$H$9$k(B.
$BJdBj(B\ref{lemma:global_minimal}, $BL?Bj(B\ref{all_are_cube}\,$B$HJQ?tJQ49$N8x<0$+$i(B
$\varDelta(E)=\pm\ep^{3n}$ ($-2\le n<2$)$B$H$7$F$h$$(B.
$c_4(E)^3-c_6(E)^2=1728\varDelta(E)$$B$@$+$i(B,
$(c_4(E),c_6(E))$$B$O(B$E_{3n}^\pm:y^2=x^3\pm1728\ep^{3n}$
$B$N(B$\calO_k$-$B@0?tE@$N=89g(B$E_{3n}^{\pm}(\calO_k)$$B$K4^$^$l$k(B.
$BA0$K=R$Y$?$h$&$K(B, $\varDelta(E)$$B$NId9f(B, $n$$B$N6v4q$O8=;~E@$G$O$o$+$i$J$$$+$i(B,
$E_{3n}^{\pm}(\calO_k)$ ($-2\le n<2$)
$B$rA4$F7h$a$k(B.
$B$7$+$70lBP0lBP1~(B
\begin{alignat*}{2}
 E_{3n}^{\pm}(\calO_{k}) &\rightarrow E_{3n+6}^\pm(\calO_{k}),\quad &
   (x,y)&\mapsto(x\ep^2,y\ep^3),
 \\
 E_3^+(\calO_{k}) &\rightarrow E_3^-(\calO_{k}),\quad &
   (x,y)&\mapsto(x'\ep^2,y'\ep^3)
\end{alignat*}
$B$,$"$k$N$G(B, $E_0^+(\calO_k)$, $E_0^-(\calO_k)$, $E_3^+(\calO_k)$
$B$N;0$D$@$17h$a$l$P==J,$G$"$k(B.

\begin{prop}
 $E_0^+(O_k)=\bigl\{(-12,0)\bigr\}$.
\end{prop}

$B$3$l$O(B$\rank E_0^+(k)=\rank(E_0^+)^{(37)}(\Q)$$B$,(B0$B$G$"$k$3$H$r<($;$P$h$$(B.
$B$3$3$K(B
\[ (E_0^+)^{(37)}:37y^2=x^3+1728. \]
$\Q$$B>eDj5A$5$l$?BJ1_6J@~$N(B$\Q$$B>e$N(Brank$B$O(B,
2-descent$B$GBgDq$N>l9g5a$a$k$3$H$,$G$-$k(B
(cf. \cite{Cremona}, \cite{Sil1}, \cite{Sil-Ta}).
$B$7$+$7(B$(E_0^+)^{(37)}$$B$N>l9g$O(B,
Tate--Shafarevich$B72$N(B($BM=A[$5$l$k(B)$B0L?t$,(B4$B$G$"$k$+$i(B,
2-descent$B$G(Brank$B$r5a$a$k$N$OBgJQ$G$"$k(B.
$B$7$+$7(B$(E_0^+)^{(37)}$$B$,5u?t>hK!$r;}$C$F$$$F(B
$L((E_0^+)^{(37)}/\Q,1)=3.1941\dots$$B$G$"$k$+$i(B,
$\rank E_0^+(k)=0$$B$,(BCoates--Wiles \cite{C-W}$B$NDjM}$h$j=>$&(B.

\bigskip
{\dg $BCm(B}.
2-descent$B$O(BCremona$B$K$h$k%W%m%0%i%`(Bmrank (rank$B$@$15a$^$k(B),
mwrank (Mordell-Weil$B72$N4pDl$b5a$^$k(B)$B$G<B9T$G$-$k(B.
SIMATH$B$K$O(B2-descent$B$H(B,
Hasse--Weil$BM=A[(B, Birch--Swinnerton-Dyer$BM=A[$r2>Dj$7$?(BManin$B$N%"%k%4%j%:%`(B
(cf. \cite{G-Z})$B$K$h$k(BMordell-Weil$B72$N4pDl$r5a$a$k%W%m%0%i%`$,$"$k(B.

\bigskip
{\dg $BCm(B}.
$E$$B$,F3<j$,$=$s$J$KBg$-$/$J$$(Bmodular$BBJ1_6J@~$N>l9g(B,
rank$B$r5a$a$k$K$O$^$:(Banalytic rank
($L$$B4X?t$N(B$s=1$$B$G$NNmE@$N0L?t(B)
$B$r7W;;$9$k$3$H$,M-1W$G$"$k(B.
Analytic rank$B$O(BSIMATH, UPECS$B$J$I$G5a$a$i$l$k(B.
SIMATH$B$K$O<c43(Bbug$B$,$"$j(B, $B$$$/$D$+$N>l9g(B
($B0lHVM_$7$$(B$(E_0^+)^{(37)}$$B$N>l9g$r4^$`(B)$B$KBP$7Ez$,JV$C$F$3$J$+$C$?(B.
PARI/GP$B$G$O(B$L(E/\Q,s)$$B$O5a$a$i$l$k$,(B,
$L^{(n)}(E/\Q,s)$ ($n\ge1$)$B$O5a$a$i$l$J$$(B.

\bigskip
$B0J2<(B$\pi=(7+\sqrt{37})/2$$B$r(B3$B$r3d$kAG85$N0l$D$H$9$k(B.

\begin{prop}
\[ E_3^+(\calO_{k})=
   \bigl\{(-12\ep,0),
          (17640-1740\sqrt{37},\pm(2074464-438480\sqrt{37}))
   \bigr\}.
\]
\end{prop}

{\dg $B>ZL@(B($B35N,(B)}.
$L:=k(\sqrt{3\ep})$$B$GJ,2r$7$F9M$($k$N$G(B,
$BI,MW$J(B$L$$B$N>pJs$r(BKASH$B$G5a$a$F$*$/(B: \par
(a) $\calO_L=\calO_k\oplus\calO_k\sqrt{3\ep}$. \par
(b) $L$$B$N4pK\C1?t7O$O(B$\ep,~\ep_1:=\ep+2\sqrt{3\ep}$$B$G$"$k(B.
    $N_{L/k}(\ep_1)=1$$B$KCm0U(B. \par
(c) $2,\pi,\pi'$$B$O(B$L$$B$G(B
    $(2)=\gerP_2^2$,~$(\pi)=\gerP_3^2$,~$(\pi')=\gerP_3^{\prime2}$
    $B$HJ,2r$9$k(B. \par
(d) $L$$B$NN`?t$O(B$2$$B$G$"$k(B. \par
\medskip
$B$3$l$r;H$&$3$H$K$h$j(B,
\[ \pm y+24\ep\sqrt{3\ep}=\ep_1^m(a+b\sqrt{3\ep})^3,~
   a,b,y\in\calO_k,~m=0,1
\]
$B$r2r$1$P$h$$$3$H$,$o$+$k(B.
$m=1$$B$N;~$K2r$,L5$$$3$H$O(B,
$\pi^2$$B$rK!$H$7$F9M$($k$J$I$9$l$PMF0W$K$o$+$k(B.

$m=0$$B$N;~(B,
$B78?t$rHf3S$7$F(B
\[
  8\ep=b(a^2+\ep b^2),~~\pm y=a(a^2+9\ep b^2)
\]
$B$,F@$i$l$k(B.
$B0l$DL\$N<0$N2r$O(B$(a,b)=(0,2)$, $(\pm 84,2\ep^{-2})$$B$N$_$G$"$j(B,
$BFs$DL\$N<0$+$i(B, $y=0$, $\pm(2074464-438480\sqrt{37})$$B$,F@$i$l$k(B.
\qed

\begin{prop}
 $E_0^-(\calO_k)$$B$O<!$N(B15$B8D$N85$+$i@.$k(B.
\begin{align*}
 &(12,0),~(16,\pm8\sqrt{37}),~(120,\pm216\sqrt{37}),
  (3376,\pm32248\sqrt{37}),\\
 &\bigl(44+4\sqrt{37},\pm(320+40\sqrt{37})\bigr),\ 
  \bigl(44-4\sqrt{37},\pm(320-40\sqrt{37})\bigr),\\
 &\bigl(572+92\sqrt{37},\pm(19040+3128\sqrt{37})\bigr),~
  \bigl(572-92\sqrt{37},\pm(19040-3128\sqrt{37})\bigr).
\end{align*}
\end{prop}

{\dg $B>ZL@(B($B35N,(B)}.
$B:#EY$O(B$L:=k(\sqrt{-3})$$B$G9M$($k$N$G(B, $L$$B$N>pJs$r$d$O$j(BKASH$B$G5a$a$F$*$/(B: \par
(a) $\calO_L=\calO_k\oplus\calO_k\zeta$\ \ ($\zeta=(1+\sqrt{-3})/2$). \par
(b) $\calO_L^\times=\langle\ep\rangle\times\langle\zeta\rangle
    \cong \Z \oplus \Z/6\Z$. \par
(c) $2,\pi,\pi'$$B$O(B$L$$B$K$*$$$F(B
    $(2)=\gerP_2\overline{\gerP}_2~(\gerP_2\ne\overline{\gerP}_2)$,~
    $(\pi)=\gerP_3^2$,~$(\pi')=\gerP_3^{\prime2}$
    $B$HJ,2r$9$k(B. \par
(d) $L$$B$N(Bideal$BN`72$O(B$\gerP_2$$B$NN`$K@8@.$5$l$k0L?t(B4$B$N=d2s72$G$"$k(B. \par
(e) $\gerP_2^4=(1+\omega-3\zeta)$.

\medskip\noindent
$B$3$l$r;H$($P(B
\[
  (\pm y+24\sqrt{-3})=\gerP_2^{a_2}\overline{\gerP}_2^{\overline a_2}\gerC^3,
  \quad
  y\in\calO_k,~(a_2,\overline a_2)=(0,0),(2,1),~\text{$\gerC$$B$O(B$L$$B$N@0(Bideal}
\]
$B$r2r$1$P$h$$$3$H$,$o$+$k(B.

Case 1: $(a_2,\overline a_2)=(0,0)$.
$L$$B$NN`?t$,(B3$B$HAG$J$3$H$+$i(B$\gerC$$B$,C19`$G$"$k$3$H$,$o$+$j(B,
\[ \pm y+24\sqrt{-3}=\ep^m\zeta^n(a+b\zeta)^3,~~
   a,b,y\in\calO_k,~m=0,\pm1,~n=0,1
\]
$B$r2r$1$P$h$$$3$H$,$o$+$k(B.
$n=1$$B$N;~$O2r$,$J$$$3$H$,$o$+$k(B.

$n=0$$B$N;~$O(B
\begin{align}
      16 &= ab(a+b),                         \label{Thue} \\
   \pm y &= \frac{\,1\,}{2}(a-b)(2a+b)(a+2b) \label{y}
\end{align}
$B$r2r$/$3$H$K$J$k(B.
(\ref{Thue})$B$N2r$O(B$(a,b)=(2,2),(2,-4),(-4,2)$$B$N$_$G$"$j(B,
(\ref{y})$B$h$j$$$:$l$N>l9g$b(B$y=0$$B$G$"$k$3$H$,$o$+$k(B.

Case 2:$(a_2,\overline a_2)=(2,1)$.
$BN>JU$K(B$(4)=(\gerP_2\overline{\gerP}_2)^2$$B$r3]$1(B,
(d), (e)$B$rMQ$$$l$P(B
\[ (4)(\pm y+24\sqrt{-3})=\gerP_2^4(\overline{\gerP}_2\gerC)^3
                         =(1+\omega-3\zeta)(\overline{\gerP}_2\gerC)^3
\]
$B$,F@$i$l(B, $L$$B$NN`?t$,(B3$B$HAG$G$"$k$3$H$h$j(B,
\[ 4(\pm y+24\sqrt{-3})=\zeta^n(1+\omega-3\zeta)(a+b\zeta)^3,
   ~a,b,y\in\calO_k,~n=0,\pm1
\]
$B$r2r$1$P$h$$$3$H$,$o$+$k(B.

$n=\pm1$$B$N;~$O2r$,L5$$$3$H$,<($;$k(B.
$n=0$$B$N;~$O(B, $B78?t$rHf3S$9$k$3$H$K$h$j(B,
\begin{align}
    &-64=a^3-(\omega-2)a^2b-(\omega+1)ab^2-b^3,  \label{Thue2}\\
    &\pm4y-96=(\omega+1)a^3+9a^2b-3(\omega-2)ab^2-(\omega+1)b^3 \label{y2}
\end{align}
$B$,F@$i$l(B,
$B8e$G8+$k$h$&$K(B(\ref{Thue2})$B$N2r$O<!$N(B21$B8D$N$_$G$"$k(B:
{\allowdisplaybreaks
\begin{align*}
 &(4,-4),~(0,4),~(-4,0), \\
 &(-3+\sqrt{37},-2\sqrt{37}),~(-2\sqrt{37},3+\sqrt{37}),~(3+\sqrt{37},-3+\sqrt{37}), \\
 &(-40-4\sqrt{37},8\sqrt{37}),~(8\sqrt{37},40-4\sqrt{37}),~(40-4\sqrt{37},-40-4\sqrt{37}), \\
 &(-2,3+\sqrt{37}),~(-1-\sqrt{37},-2),~(3+\sqrt{37},-1-\sqrt{37}), \\
 &(-3+\sqrt{37},2),~(1-\sqrt{37},-3+\sqrt{37}),~(2,1-\sqrt{37}), \\
 &(-19-3\sqrt{37},16+2\sqrt{37}),~(16+2\sqrt{37},3+\sqrt{37}),~(3+\sqrt{37},-19-3\sqrt{37}), \\
 &(-16+2\sqrt{37},19-3\sqrt{37}),~(-3+\sqrt{37},-16+2\sqrt{37}),~(19-3\sqrt{37},-3+\sqrt{37}).
\end{align*}
}
$B$3$l$i$r(B(\ref{y2})$B$KBeF~$7$F(B,
$BL?Bj$K=R$Y$?$b$N$N$&$A(B$y=0$$B0J30$NA4$F$NCM$,F@$i$l$k(B.
\qed

\bigskip
{\dg $BCm(B}.
$E_0^-(k)=(E_0^-)^{(37)}(\Q)$$B$N(Brank$B$O(B2$B$G(B,
$B$3$l$O(B2-descent$B$G4JC1$K5a$^$k(B.

\bigskip
$B$3$l$G(B$E_{3n}^\pm(\calO_k)$ ($-2\le n<2$)$B$,A4$F7h$^$C$?(B.
$BB($A(B
$k$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$N(B$c_4,c_6$$B$N8uJd$,=PB7$C$?$o$1$G$"$k(B.
$(x,y)\in E_{3n}^\pm(\calO_k)$$B$,(B$k$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$N(B
global minimal model$B$N(B$c_4,c_6$$B$K@.$l$k$3$H$H(B,
$Y^2=X^3-27xX-54y$$B$N(B$k$$B>e$NF3<j$,(B(1)$B$G$"$k$3$H$OF1CM$G$"$k(B.
$B<B:]$KF3<j$r7W;;$9$k$3$H$K$h$j(B,
$c_4,c_6$$B$K@.$l$k$N$O(B
\[ (16\ep^{-2},\,-8\sqrt{37}\ep^{-3})~,~(3376\ep^{-2},\,32248\sqrt{37}\ep^{-3})
   \in E_{-6}^-(\calO_k)
\]
$B$NFs$D$N$_$G$"$k$3$H$,$o$+$k(B.
$j$-invariant$B$rHf3S$9$l$P(B, $BA0<T$,(B37A1$B$K(B, $B8e<T$,(B37A2$B$KBP1~$9$k$3$H$,$o$+$k(B.
$B$h$C$F<!$,F@$i$l$?(B:

\begin{th}
$\Q(\sqrt{37})$$B>e;j$k=j(Bgood reduction$B$r;}$DBJ1_6J@~$O(B
37A1, 37A2$B$N$_$G$"$k(B. \qed
\end{th}

$BFs<!BN>e$NBJ1_6J@~$NF3<j$O(BSIMATH$B$G5a$^$k(B($B:G?7$N(Bversion 4.1$B$K$b<c43(Bbug$B$,$"$k(B).
$BG$0U$NBe?tBN>e$NBJ1_6J@~$NF3<j$OG_3@;a$,(BPARI/GP$B$G:n@.$7$?%W%m%0%i%`$G5a$^$k(B
(cf. \cite{Ume2}).

\section{$\Q(\protect\sqrt{37})$$B>e$N(BThue$BJ}Dx<0(B}
$B$3$N@a$G$O(B(\ref{Thue2})$B$r2r$/(B.
$k=\Q(\sqrt{37})$$B$H$9$k(B.

$a,b\in\calO_k$$B$r(B(\ref{Thue2})$B$N2r$H$9$k(B.
$A=-a-(\omega+2)b$$B$,(B4$B$G3d$l$k$3$H$H(B$b$$B$,(B2$B$G3d$l$k$3$H$OD>$A$K$o$+$k(B.
$A=4X$, $b=2Y$, $X,Y\in\calO_k$$B$H$*$/$H(B,
\begin{equation} \label{Thue_3}
  X^3+2(\omega+1)X^2Y+4(\omega+3)XY^2+2(2\omega+5)Y^3=1
\end{equation}
$B$,F@$i$l$k(B.
(\ref{Thue_3})$B$N$h$&$JFs<!BN>e$N(BThue$BJ}Dx<0$O(B
de Weger$B$,(B\cite{dW}$B$K$*$$$F2r$$$?$b$N$,(B($BI.<T$NCN$k8B$j(B)$BM#0l$G$"$k(B.
$B$=$3$G$N>ZL@$HF1MM$NJ}K!$G<!$r<($9(B:

\begin{prop}
(\ref{Thue_3})$B$rK~$?$9(B$(X,Y)\in\calO_k\times\calO_k$$B$O(B
$B0J2<$N(B21$B8D$N$_$G$"$k(B:
{\allowdisplaybreaks
\begin{align*}
&(-2-9\omega,22-4\omega),~(-23-8\omega,-4+8\omega),~
 (25+17\omega,-18-4\omega), \\
&(21+8\omega,-8-3\omega),~(-9-3\omega,1+\omega),~(-12-5\omega,7+2\omega), \\
&(9+2\omega,1-2\omega),~(-3-\omega,-2 +\omega),~(-6-\omega,1 +\omega), \\
&(-5-2\omega,1+\omega),~(1+\omega,-1),~(4+\omega,-\omega), \\
&(-2-\omega,2),~(1,0),~(1+\omega,-2), \\
&(3+\omega,1-\omega),~(-\omega,1),~(-3,-2+\omega), \\
&(7-2\omega,11-3\omega),~(1+\omega,-9+2\omega),~(-8+\omega,-2+\omega).
\end{align*}
}
\end{prop}

{\dg $B>ZL@(B($B35N,(B).}
$F(X,Y)$$B$r(B(\ref{Thue_3})$B$N:8JU(B,
$\theta$$B$rB?9`<0(B$F(X,1)$$B$N:,$N0l$D$H$7(B,
$L=\Q(\theta)$$B$H$*$/(B.
$B$9$k$H(B$k\subset L$$B$G$"$j(B, $[L:\Q]=6$, $\calO_L=\Z[\xi]$$B$G$"$k(B.
$B$3$3$K(B$\xi=(12+18\theta-4\theta^3-\theta^4)/20$.
$L/\Q$$B$O(BGalois$B3HBg$G(B, Galois$B72$O(B
$\langle\sigma,\tau\rangle$,
\begin{align*}
 \sigma(\xi)&=-14-6\xi+49\xi^2+9\xi^3-28\xi^4-6\xi^5, \\
 \tau(\xi)  &=-1-3\xi+5\xi^2+4\xi^3-4\xi^4-\xi^5,
\end{align*}
$B$G$"$k(B.
$\sigma$, $\tau$$B$O(B$\sigma^3=1$, $\tau^2=1$, $\sigma\tau=\tau\sigma^2$$B$rK~$?$9(B.
$BB($A(B$\Gal(L/\Q)$$B$O(B3$B<!BP>N72$G$"$k(B.
$L$$B$N(B1$BAH$N4pK\C1?t7O$O<!$GM?$($i$l$k(B:
\begin{align*}
 \ep_1&=-\xi, \\
 \ep_2&=-5-4\xi+18\xi^2+5\xi^3-9\xi^4-2\xi^5, \\
 \ep_3&=-6-8\xi+23\xi^2+9\xi^3-13\xi^4-3\xi^5, \\
 \ep_4&=1+3\xi-5\xi^2-4\xi^3+4\xi^4+\xi^5, \\
 \ep_5&=-16-15\xi+63\xi^2+18\xi^3-36\xi^4-8\xi^5.
\end{align*}
$N_{L/k}(\ep_i)=1$ ($i=1,2,3,4$), $N_{L/k}(\ep_5)=\ep$$B$,$o$+$k(B.
(KASH$B$rMQ$$$FF@$i$l$?4pK\C1?t7O$rMQ$$$F$3$&$J$k$h$&$K$7$?$N$G$"$k(B.)

(\ref{Thue_3})$B$O(B$N_{L/k}(X-Y\theta)=1$$B$HF1CM$G$"$k$+$i(B,
$\eta:=X-Y\theta=\ep_1^{a_1}\ep_2^{a_2}\ep_3^{a_3}\ep_4^{a_4}$
($a_1,\dots,a_4\in\Z$)$B$G$"$k(B.
$X,Y$$B$r>C5n$7$F(B
\[  \bigl(\sigma(\theta)-\sigma^2(\theta)\bigr)\eta
   +\bigl(\sigma^2(\theta)-\theta\bigr)\sigma(\eta)
   +\bigl(\theta-\sigma(\theta)\bigr)\sigma^2(\eta)=0,
\]
$B$h$C$F(B
\[
    \frac{~\theta-\sigma^2(\theta)~}{\theta-\sigma(\theta)}
    \cdot
    \frac{\sigma(\eta)}{~\sigma^2(\eta)~}-1
  =-\frac{~\sigma(\theta)-\sigma^2(\theta)~}{\sigma(\theta)-\theta}
    \cdot
    \frac{\eta}{\,\sigma^2(\eta)\,}.
\]
$B$3$l$O<!$HF1CM$G$"$k(B:
\begin{equation} \label{Siegel}
    -\ep_1^{b_1}\ep_2^{b_2}\ep_3^{b_3}\ep_4^{b_4}-1
   = \ep_1^{d_1}\ep_2^{d_2}\ep_3^{d_3}\ep_4^{d_4}.
\end{equation}
$B$3$3$K(B
\begin{align*}
  b_1&=a_1+2a_3,~b_2=a_2+2a_4-1,~b_3=-2a_1-a_3+1,~b_4=-2a_2-a_4, \\
  d_1&=-b_3,~d_2=-b_4,~d_3=b_1+b_3,~d_4=b_2+b_4.
\end{align*}
Thue$BJ}Dx<0$r2r$/;~$K$$$D$b$d$k$h$&$K(B
(cf.$\;$\cite{Kida2}, \cite{T-dW}, \cite{dW}),
linear forms in the logarithms
\[ \varLambda_i=\sum_{j=1}^4 b_j\log|\ep_j^{(i)}|=
   \begin{cases}
      \log
      \left|
        \dfrac{\,\theta^{(i)}-\sigma^2(\theta^{(i)})\,}%
              {\theta^{(i)}-\sigma(\theta^{(i)})}
        {\cdot}
        \dfrac{\sigma(\eta^{(i)})}{\,\sigma^2(\eta^{(i)})\,}
      \right| & (1\le i\le3),  \\
   & \\
      \log
      \left|
         \dfrac{\theta^{(i)}-\sigma(\theta^{(i)})}%
               {\,\theta^{(i)}-\sigma^2(\theta^{(i)})\,}
         {\cdot}
         \dfrac{\,\sigma^2(\eta^{(i)})\,}{\sigma(\eta^{(i)})}
      \right| & (4\le i\le6)
   \end{cases}
\]
$B$rI>2A$9$k(B.
$i_0\in\{1,\dots,6\}$$B$r(B
$|\eta^{(i_0)}|=\min\limits_{1\le i\le6}\{|\eta^{(i)}|\}$$B$G7h$a$k(B.
\cite{dW}$B$N5DO@$HF1MM$K$7$F(B, $B\ge100$$B$J$i(B
\begin{equation}  
 |\varLambda_{i_0}|<4.1069\exp(-0.24457B) \label{ub1}
\end{equation}
$B$,@.$jN)$D$3$H$,$o$+$k(B.
$B$3$l$rF@$k$N$K(B$\theta$, $\ep_i$ ($i=1,2,3,4$)$B$N6&Lr$N6a;wCM$r;H$&$,(B,
$B$=$l$O(BKASH$B$G5a$a$i$l$k(B.

$B<!$K(B, Baker$BM}O@$rMQ$$$F(B$|\varLambda_{i_0}|$$B$N(Blower bound$B$r5a$a$k(B.
\cite{dW}$B$N>l9g$HF1$8$/(B, $i_0=1$$B$H$7$F0lHL@-$r<:$o$J$$(B.
\cite{B-W}$B$NDjM}$h$j(B
\begin{equation} \label{lb}
 \log|\varLambda_1|>-4.1810\times10^{18}\log(B)
\end{equation}
$B$,F@$i$l$k(B.
$B$3$N;~$K(B$\ep_i$ ($i=1,2,3,4$)$B$N(Bheight$B$,I,MW$K$J$k$,(B, $B$=$l$b(BKASH$B$G5a$a$i$l$k(B.

(\ref{ub1}), (\ref{lb})$B$r$"$o$;$F(B, $B\le1.5142\times10^{21}$$B$,F@$i$l$k(B.

$B$3$N(Bupper bound$B$OBg$-$9$.$k$N$G(B,
$BE,Ev$JBg$-$5$^$G2<$2$kI,MW$,$"$k(B.
$B$=$N$?$a$K<!$NJdBj(B(\cite{T-dW}$B$N(BProposition 3.1)$B$r;H$&(B

\begin{lemma} \label{new_ub}
$\mu_1,\dots,\mu_n$$B$rM?$($i$l$?<B?t$H$9$k(B.
$b_1,\dots,b_n\in\Z$$B$rJQ?t$H$7(B,
$\varLambda=\sum\limits_{i=1}^n b_i\mu_i$$B$H$*$/(B.
$K_1,K_2,K_3$$B$rM?$($i$l$?@5$N<B?t$H$7(B,
$b_1,\dots,b_n$$B$,(B
\begin{equation} \label{inequ}
 |\varLambda|<K_1\exp(-K_2 B),~~B:=\max\bigl\{|b_1|,\dots,|b_n|\bigr\}<K_3
\end{equation}
$B$rK~$?$9$H$9$k(B.
$B==J,Bg$-$J<B?t(B$c_0$$B$KBP$7(B,
\[
  {\cal A}=
  \begin{pmatrix}
         1     &         &                & 0\\
               & \ddots  &                & \vdots \\
               &         &        1       & 0 \cr
    [c_0\mu_1] &  \dots  & [c_0\mu_{n-1}] & [c_0\mu_n]
  \end{pmatrix}.
\]
$B$NNs$G@8@.$5$l$k(Blattice $\varGamma$$B$r9M$($k(B.
$B$3$3$K(B
\[ [x]=
   \begin{cases}
     \lfloor x \rfloor & \text{if $x\ge0$,} \\
     \lceil  x \rceil  & \text{if $x<0$,}
   \end{cases}
\]
$BB($A(B$[\cdot]$$B$O(B0$B$K8~$+$C$F$N@Z$j<N$F$G$"$k(B.
$(\bold{b}_1,\cdots,\bold{b}_n)$$B$r(B$\varGamma$$B$N(BLLL-reduced basis$B$H$9$k(B.
$B$b$7(B$|\bold b_1|>\sqrt{(n^2+n-1)2^{n-1}}K_3$$B$,@.$jN)$D$J$i$P(B,
(\ref{inequ})$B$NA4$F$N2r$O(B
\[
 B<
 \frac{\,\log(c_0K_1)-\log(\sqrt{2^{1-n}|\bold b_1|^2-(n-1)K_3^2}-nK_3)\,}{K_2}
\]
$B$rK~$?$9(B. \qed
\end{lemma}

$c_0=10^{100}$$B$H$7(B
$B9TNs(B${\cal A}$$B$K(BLLL-reduction algorithm$B$rE,MQ$9$k(B($B$3$3$G$O(BPARI/GP$B$r;H$C$?(B)$B$H(B,
\[ {\cal B}=\bigl(\bold b_1 \ \bold b_2 \  \bold b_3 \  \bold b_4 \bigr), \]
\begin{alignat*}{2}
 \bold b_1&=\left(
\begin{array}{r}
   525766899856084740716174 \\
  3846389868324456104273427 \\
 -1244186664511728113718131 \\
  -395108746616005504770747 
\end{array}
\right)\!,~ &
\bold b_2&=\left(
\begin{array}{r}
 -3580522850813688135299104 \\
  -341447815688279270973156 \\
  1727813860773260342514675 \\
  3246721051937534783355873
\end{array}
\right)\!,\\
\bold b_3&=\left(
\begin{array}{r}
  4072674279999564495273127 \\
  2692442070527295763521844 \\
  7820253876673256339974486 \\
 -2851019503830648230431094 \\
\end{array}
\right)\!,~ 
&
\bold b_4&=\left(
\begin{array}{r}
   -7825402845303750147594994 \\
   -1547312398964229893583459 \\
    -529196120215387679117837 \\
  -10620598711855356914189251
\end{array}
\right)
\end{alignat*}
$B$,F@$i$l$k(B.
$|\bold b_1|=4.096\dots\times10^{24}>
 \sqrt{(n^2+n-1)2^{n-1}}K_3=1.866\times10^{22}$$B$G$"$k$+$i(B,
$BJdBj(B\ref{new_ub}\,$B$h$j(B$B\le719$$B$,$o$+$k(B.

$c_0=10^{18}$$B$H<h$k(B.
$B:F$S(B${\cal A}$$B$K(BLLL-reduction algorithm$B$rE,MQ$9$k$H(B,
\[ (\bold b_1\ \bold b_2\ \bold b_3\ \bold b_4)=\left(
\begin{array}{rrrr}
 -291 &  -1300 & 23101 &  13586 \\
 2046 &   2852 &  6305 & -24467 \\
19892 &   7913 &  5062 &  -1315 \\
  285 & -18603 & -7284 &  -5310 
\end{array}
\right).
\]
$|\bold b_1|=2.000\dots\times10^4>\sqrt{(n^2+n-1)2^{n-1}}K_3=8.874\times10^3$,
$B$G$"$k$+$i(B, $B\le141$$B$,F@$i$l$k(B.

$B;D$C$F$$$k$N$O(B, $B\le141$$B$NHO0O$G(B(\ref{Siegel})$B$N2r$rC5$9$3$H$G$"$k(B.
$B@5D>$K$d$k$H;~4V$,$+$+$k$N$G(B, \cite{dW}$B$HF1MM$N9)IW$r$9$k(B.
$B$=$&$9$l$P(B, C$B8@8l$G=q$$$?(Bprogram$B$r(BSparc station SS4$B$GAv$i$;$l$P(B,
$BA4(B21$B8D$N2r$,(B15$BJ,$G8+$D$+$k(B.
$B3F2r(B$(a_1,a_2,a_3,a_4)$$B$KBP$7(B,
$\ep_1^{a_1}\ep_2^{a_2}\ep_3^{a_3}\ep_4^{a_4}$$B$,(B$X-Y\theta$$B$N7A$r$7$F$$$k$3$H$,(B
$B$d$O$j(BKASH$B$G3N$+$a$i$l$k(B
(KASH$B$O(Brelative extension$B$b07$($k$N$G$"$k(B).
$BA4$F$N2r$O<!$NI=$NDL$j$G$"$k(B:

{\small
\begin{center}
\makeatletter
\def\@captype{table}
\makeatother
\begin{tabular}{rrrrrrrrrr}
$a_1$ & $a_2$ & $a_3$ & $a_4$ & $b_1$ & $b_2$ & $b_3$ & $b_4$ &  $X$ & $Y$ \\
\hline
$-3$  & $-4$  &  $-1$ &    5  & \quad $-5$  &  5    &  8    &  3    & \quad$-2-9\omega$  & $22-4\omega$ \\
  0   &  4    &    4  &    0  &   8   &  3    & $-3$  & $-8$  & $-23-8\omega$ & $-4+8\omega$  \\
  5   & $-1$  &  $-4$ &  $-3$ & $-3$  & $-8$  & $-5$  & 5     & $25+17\omega$ & $-18-4\omega$ \\
  4   & $-1$  &  $-4$ &    1  & $-4$  & 0     & $-3$  & 1     & $21+8\omega$ & $-8-3\omega$ \\
$-3$  & 0     &     0 &    1  & $-3$  &     1 &     7 &  $-1$ & $-9-3\omega$ & $1+\omega$ \\
    1 &     0 &     3 &     0 &    7  &  $-1$ &  $-4$ &     0 & $-12-5\omega$ & $7+2\omega$ \\
    3 &  $-3$ &  $-3$ &     3 &  $-3$ &     2 &  $-2$ &     3 & $9+2\omega$ & $1-2\omega$ \\
 $-2$ &     2 &     0 &     1 &  $-2$ &     3 &     5 &  $-5$ & $-3-\omega$ & $-2+\omega$ \\
    1 &     0 &     2 &  $-2$ &    5  &  $-5$ &  $-3$ &     2 & $-6-\omega$ & $1+\omega$ \\
    2 &     0 &  $-2$ &     1 &  $-2$ &     1 &  $-1$ &  $-1$ & $-5-2\omega$ & $1+\omega$ \\
 $-1$ &     0 &     0 &     0 &  $-1$ &  $-1$ &     3 &     0 & $1+\omega$ & $-1$ \\
    1 &  $-1$ &     1 &     1 &     3 &     0 &  $-2$ &     1 & $4 + \omega$ & $-\omega$ \\
    1 &     0 &  $-1$ &     1 &  $-1$ &     1 &     0 &  $-1$ & $-2 - \omega$ & $2$ \\
    0 &     0 &     0 &     0 &      0 & $-1$ &     1 &     0 & 1        & 0 \\
    1 &  $-1$ &     0 &     1 &      1 &    0 &  $-1$ &     1 & $1+\omega$ & $-2$ \\
    1 &     1 &  $-1$ &     1 &   $-1$ &    2 &     0 &  $-3$ & $3+\omega$ & $1-\omega$ \\
    0 &     0 &     0 &  $-1$ &      0 & $-3$ &     1 &     1 & $-\omega$  & 1  \\
    1 &  $-2$ &     0 &     2 &      1 &   1  &  $-1$ &     2 & $-3$ & $-2+\omega$ \\
    1 &  $-4$ &  $-1$ &     4 &   $-1$ &    3 &     0 &     4 & $7-2\omega$ & $11-3\omega$ \\
    0 &     3 &     0 &     1 &      0 &    4 &     1 & $-7$ & $1+\omega$ & $-9+2\omega$ \\
    1 &     0 &     0 &  $-3$ &      1 & $-7$ &  $-1$ &    3 & $-8+\omega$ & $-2+\omega$ \\
\hline
\end{tabular}
\caption{(\ref{Thue_3}), (\ref{Siegel})$B$NA4$F$N2r(B} \label{sol}
\end{center}
}

\section{Appendix 1: Free-ware}
$BK\J8Cf$K=P$F$-$?(Bfree-ware$B$O(B, $B$=$N8e$K=q$$$?(BURL$B$+$iF~<j2DG=(B.

\begin{tabular}{ll}
KASH          & ftp://math.tu-berlin.de/pub/algebra/Kant/Kash \\
mrank, mwrank & ftp://euclid.ex.ac.uk/pub/cremona \\
PARI/GP       & ftp://megrez.math.u-bordeaux.fr/pub/pari \\
SIMATH        & ftp://ftp.math.uni-sb.de/pub/simath \\
Upecs         & ftp://math.mcgill.ca/pub/upecs
\end{tabular}

\medskip\noindent
Upecs$B$O(Bmemory 640KB$B$N%Q%=%3%s$G$bF0$/$N$G(B, $B=PMh$k$3$H$O8B$i$l$F$$$k$,(B,
$B$D$$:G6a$^$G(B10$BG/0J>eA0$N%Q%=%3%s$r;H$C$F$$$?I.<T$K$H$C$F$O=EJu$G$"$C$?(B.
\section{Appendix 2: Table}
$B$3$3$G$O(B$k=\Q(\sqrt m)$ ($m=6,7,14,29,33,37,41,65$)$B>e;j$k=j(Bgood reduction$B$r;}$D(B
$BA4BJ1_6J@~$NI=$rM?$($k(B.
$k$-isogeny classes$B$K$bJ,$1$F$"$k(B.
$BI=$K$O3F6J@~$N%G!<%?$,<!$N$h$&$KM?$($i$l$F$$$k(B:

\bigskip
(1) $mXi$$B$N7A$N(Bcode.
    $m$$B$O6J@~$,(B$k=\Q(\sqrt m)$$B>eDj5A$5$l$F$$$k$3$H$r0UL#$9$k(B.
    $X=A,B$$B$O(B$k$-isogeny class$B$rI=$9(B.
    (isogeny class$B$,Fs$D$"$k$N$O(B$m=65$$B$N;~$@$1$@$,(B.)
    $i$$B$OC1$J$k=gHV$G$"$k(B.
    $m=6,7,14,41,65$$B$N;~$O(B,
    Comalada \cite{Com}$B$N(Bcode$B$b=q$$$F$*$$$?(B. \par
(2) $BDj5AJ}Dx<0$N78?t(B$a_1,a_2,a_3,a_4,a_6$. \par
(3) $BH=JL<0(B. \par
(4) $j$-invariant. \par
(5) Torsion subgroup$B$N9=B$(B.
    $C_n$$B$O0L?t(B$n$$B$N=d2s72$N$3$H$H$9$k(B. \par
(6) Isogeny graph.
    $k$$B>eDj5A$5$l$?BJ1_6J@~(B$E$, $\bar E$, $B5Z$SAG?t(B$p$$B$KBP$7(B,
\[
  E~\raisebox{.5ex}{$\overset{p}{\hbox to 2em{\hrulefill}}$}~\bar E
\]
$B$O(B$E$$B$H(B$\bar E$$B$,(B$k$$B>e(B$p$-isogenous$B$G$"$k$3$H$r0UL#$9$k(B.

\bigskip
Torsion subgroup$B$N9=B$$O(B,
$E(k)_{\mathrm{tors}}$$B$+$i(Bgood prime $\gerp$$B$G$N(Breduction$B$N(B
$\calO_k/\gerp$-$BM-M}E@$N72(B$E_{\gerp}(\calO_k/\gerp)$$B$X$N(B
$BC1<M=`F17?$,$"$k$3$H$rMQ$$$l$P$"$kDxEY$7$\$l$k(B.
$B$7$+$7F1$8(B$k$-isogeny class$B$KB0$9$k(B$E$, $\bar E$$B$KBP$7$F$O(B,
$E_{\gerp}(\calO_k/\gerp)=\bar E_{\gerp}(\calO_k/\gerp)$$B$,(B
$BA4$F$N(Bgood prime $\gerp$$B$KBP$7$F@.$jN)$D$N$G(B,
$B$3$l$@$1$G$O$@$a$G$"$k(B.
$B:G=*7k2L$r=P$9$K$O(B, $n$$BEyJ,B?9`<0(B$\psi_n$$B$N(B$k$$B$G$NJ,2r$rMQ$$$k(B,
\cite{MSZ}$B$NI=(B
($BFs<!BN(B$k$$B>eDj5A$5$l$?BJ1_6J@~(B$E$$B$G(B,
$j(E)\in\calO_k$$B$G$"$j(B$E(k)_{\mathrm{tors}}$$B$,(B
$\Z/2\Z$, $\Z/3\Z$, $\Z/2\Z\oplus\Z/2\Z$$B$G$J$$$b$N$OM-8B8D$7$+$J$/(B,
$B$=$NI=$,M?$($i$l$F$$$k(B)
$B$NCf$+$iF3<j(B(1)$B$N$b$N$r8+$D$1$@$9$J$I$9$l$P$h$$(B.

$k$-isogeny class$B$KJ,$1$k$K$O(B,
torsion point$B$r5a$a$k(B,
$\psi_n$$B$NJ,2r$r5a$a$k(B,
$B$J$I$r$7$?$N$A(BV\'elu$B$N8x<0(B(\cite{Velu})$B$r;H$($P$h$$(B.
$B$=$&$O$$$C$F$b(B$\psi_7$, $\psi_{11}$$B$J$I$NJ,2r$O;~4V$,$+$+$k$N$G(B,
$\Q$-curve$B$K4X$9$k7k2L(B(\cite{Hase})$B$r;H$&(B,
$j\in\Z$$B$N;~$O(B\cite{Cremona}$B$NI=$N6J@~$N(Btwist$B$K$J$C$F$$$k$+$I$&$+3N$+$a$k(B,
$B$J$I$r$9$k(B.
$B$3$NJ}K!$K$h$j(B, $BI=Cf$N(B7-isogeny$B$H(B11-isogeny$B$NB8:_$r3N$+$a$?(B.

\newpage
{\footnotesize

\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
Code & $a_1$ & $a_2$ & $a_3$ & $a_4$ & $a_6$ & $\varDelta$ & $j$ & tors \\
\hline
6A1 $\phantom{'}(E_1)$ & $\phantom{-}\sqrt6$ & $-2-\sqrt6$ & $-1$ & 0 & 0 & $\ep^3$ & 8000 & $C_6$ \\
6A1$'$ ($E_2$) & $-\sqrt6$ & $-2+\sqrt6$ & $-1$ & 0 & 0 & $\ep^{\prime3}$ & 8000 & $C_6$\\
6A2 $\phantom{'}(E_3)$ & $\phantom{-}\sqrt6$ & $1-\sqrt6$ & $1+\sqrt6$ & $9-34\ep$ & $-1122-459\sqrt6$ & $\ep$ & $64(4\ep^4+1)^3\ep^{\prime4}$ & $C_2$ \\
6A2$'$ $(E_4$) & $-\sqrt6$ & $1+\sqrt6$ & $1-\sqrt6$ & $9-34\ep'$ & $-1122+459\sqrt6$ & $\ep'$ & $64(4\ep^{\prime4}+1)^3\ep^4$ & $C_2$\\
6A3 $\phantom{'}(E_5)$ & $\phantom{-}\sqrt6$ & $2+\ep'$ & $3-\sqrt6$ & $-7+3\sqrt6$ & 0 & $\ep^{\prime5}$ & $64(4\ep^4+1)^3\ep^{\prime4}$ & $C_6$ \\
6A3$'$ $(E_6)$ & $-\sqrt6$ & $2+\ep$ & $3+\sqrt6$ & $-7-3\sqrt6$ & 0 & $\ep^5$ & $64(4\ep^{\prime4}+1)^3\ep^4$ & $C_6$ \\
\hline
\end{tabular}

\bigskip
\setlength{\unitlength}{1em}
\begin{picture}(7,8)(-1,4)
  \put(0,10){6A2}
  \put(2,10.3){\line(1,0){3.7}}
  \put(6,10){6A1}
  \put(7.9,10.3){\line(1,0){3.9}}
  \put(12,10){6A3}

  \put(.9,6){\line(0,1){3.6}}
  \put(6.9,6){\line(0,1){3.6}}
  \put(12.9,6){\line(0,1){3.6}}

  \put(0,5){6A2$'$}
  \put(2,5.3){\line(1,0){3.7}}
  \put(6,5){6A1$'$}
  \put(8.1,5.3){\line(1,0){3.7}}
  \put(12,5){6A3$'$}

  \put(3.5,5.7){3}  \put(3.5,10.7){3}
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  \put(0,7.5){2}  \put(6,7.5){2} \put(12,7.5){2} 
\end{picture}

\bigskip\bigskip
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
\hfil Code & $a_1$ & $a_2$ & $a_3$ & $a_4$ & $a_6$ & $\varDelta$ & $j$ & tors \\
\hline
7A1 $\phantom{'}(E_7)$ & 1 & $4\ep$ & 0 & $\ep$ & 0 & $\ep^6$ & $255^3$  & $C_2\times C_2$ \\
7A1$'$ $(E_8)$ & 1 & $4\ep'$ & 0 & $\ep'$ & 0 & $\ep^{\prime6}$ & $255^3$  & $C_2\times C_2$ \\
7A2 $\phantom{'}(E_9)$ & 1 & $4\ep$ & 0 & $6\ep-80\ep^2$ & $-3044+48513\ep$ & $\ep^3$ & $(256\ep^2+\ep')^3$ & $C_4$ \\
7A2$'$ $(E_{10})$& 1 & $4\ep'$ & 0 & $6\ep'-80\ep^{\prime2}$ & $-3044+48513\ep'$ & $\ep^{-3}$ & $(256\ep^{\prime2}+\ep)^3$ & $C_4$ \\
7A3 $\phantom{'}(E_{14})$& $\ep'$ & $-2\ep'$ & 0 & $\ep^{\prime2}$ & 0 & $-\ep^{\prime6}$ & $-15^3$ & $C_4$ \\
7A3$'$ $(E_{13})$ & $\ep$ & $-2\ep$ & 0 & $\ep^2$ & 0 & $-\ep^6$ & $-15^3$ & $C_4$ \\
7A4 $\phantom{'}(E_{12})$ & 1 & $4\ep$ & 0 & $-4\ep$ & $-\ep^3-2\ep$ & $\ep^{\prime9}$ & $(256\ep^{\prime2}+\ep)^3$ & $C_2$ \\
7A4$'$ $(E_{11})$  & 1 & $4\ep'$ & 0 & $-4\ep'$ & $-\ep^{\prime3}-2\ep'$ & $\ep^9$ & $(256\ep^2+\ep')^3$ & $C_2$ \\
\hline
\end{tabular}

\bigskip
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  \put(0,10){7A2}
  \put(1.9,10.3){\line(1,0){3.8}}
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  \put(12,10){7A3}

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  \put(12.8,6){\line(0,1){3.6}}

  \put(0,5){7A2$'$}
  \put(2.1,5.3){\line(1,0){3.6}}
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\bigskip\bigskip
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
\hfil Code & $a_1$ & $a_2$ & $a_3$ & $a_4$ & $a_6$ & $\varDelta$ & $j$ & tors \\
\hline
14A1 $\phantom{'}(E_{15})$ & $1+\sqrt{14}$ & $-9-2\sqrt{14}$ & 0 & $\ep$ & 0 & $-\ep^3$ & $-15^3$ & $C_2$ \\
14A1$'$ $(E_{16})$ & $1-\sqrt{14}$ & $-9+2\sqrt{14}$ & 0 & $\ep'$ & 0 & $-\ep^{\prime3}$ & $-15^3$ & $C_2$ \\
14A2 $\phantom{'}(E_{17})$ & $1+\sqrt{14}$ & $-9-2\sqrt{14}$ & 0 & $-4\ep$ & $651+174\sqrt{14}$ & $\ep^3$ & $255^3$ & $C_2$ \\
14A2$'$ $(E_{18})$ & $1-\sqrt{14}$ & $-9+2\sqrt{14}$ & 0 & $-4\ep'$ & $651-174\sqrt{14}$ & $\ep^{\prime3}$ & $255^3$ & $C_2$ \\
\hline
\end{tabular}

\bigskip
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  \put(8.5,2.5){2}
\end{picture}

\newpage
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
Code & $a_1$ & $a_2$ & $a_3$ & $a_4$ & $a_6$ & $\varDelta$ & $j$ & tors \\
\hline
29A1 & 1 & 0 & $\ep^2$ & 0 & 0 & $-\ep^{10}$ & $(5\ep-2)^3\ep^{\prime4}$ & $C_3$ \\ 
29A1$'$ & 1 & 0 & $\ep^{-2}$ & 0 & 0 & $-\ep^{\prime10}$ & $(5\ep'-2)^3\ep^4$ & $C_3$ \\ 
29A2 & 1 & 0 & $\ep^2$ & $-5\ep^2$ & $-(\ep^2+7\ep^4)$ & $-\ep^{14}$ & $-(1+216\ep^2)^3\ep^{\prime14}$ & 1 \\
29A2$'$ & 1 & 0 & $\ep^{\prime2}$ & $-5\ep^{\prime2}$ & $-(\ep^{\prime2}+7\ep^{\prime4})$ & $-\ep^{\prime14}$ & $-(1+216\ep^{\prime2})^3\ep^{14}$ & 1 \\
\hline
\end{tabular}

\bigskip
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  \put(4.3,5.7){5}
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  \put(8.5,2.5){3}
\end{picture}

\bigskip\bigskip
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
Code & $a_1$ & $a_2$ & $a_3$ & $a_4$ & $a_6$ & $\varDelta$ & $j$ & tors \\
\hline
33A1 & $(\ep-3)/4$ & 0 & $\ep$ & 0 & 0 & $-\ep^3$ & $-2^{15}$ & $C_3$ \\
33A1$'$ & $(\ep'-3)/4$ & 0 & $\ep'$ & 0 & 0 & $-\ep^{\prime3}$ & $-2^{15}$ & $C_3$ \\
33A2 & $(\ep-3)/4$ & 0 & $\ep$ & $(5-215\ep)/4$ & $34-1563\ep$ & $-\ep$ & $-(5+\sqrt{33})^3(243\ep-1)^3\ep'$ & 1 \\
33A2$'$ & $(\ep'-3)/4$ & 0 & $\ep'$ & $(5-215\ep')/4$ & $34-1563\ep'$ & $-\ep'$ & $-(5-\sqrt{33})^3(243\ep'-1)^3\ep$ & 1 \\
33A3 & $(\ep-3)/4$ & 0 & $\ep$ & $(15\ep-5)/4$ & $(127\ep-1)/4$ & $-\ep^5$ & $-(5-\sqrt{33})^3(243\ep'-1)^3\ep$ & 1 \\
33A3$'$ & $(\ep'-3)/4$ & 0 & $\ep'$ & $(15\ep'-5)/4$ & $(127\ep'-1)/4$ & $-\ep^{\prime5}$ & $-(5+\sqrt{33})^3(243\ep-1)^3\ep'$ & 1 \\
\hline
\end{tabular}

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\bigskip\bigskip
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
Code & $a_1$ & $a_2$ & $a_3$ & $a_4$ & $a_6$ & $\varDelta$ & $j$ & tors \\
\hline
37A1 & 0 & $(3\ep+1)/2$ & $-\ep$ & $(11\ep+1)/2$ & 0 & $\ep^6$ & $2^{12}$ & $C_5$ \\
37A2 & 0 & $(3\ep+1)/2$ & $-\ep$ & $-(1669\ep+139)/2$ & $-7(5449\ep+451)$ & $\ep^6$ & $3376^3$ & 1 \\
\hline
\end{tabular}

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\bigskip\bigskip
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
\hfil Code & $a_1$ & $a_2$ & $a_3$ & $a_4$ & $a_6$ & $\varDelta$ & $j$ & tors \\
\hline
41A1 $\phantom{'}(E_{23})$& 1 & 0 & 0 & $-\ep$ & 0 & $\ep^4$ & $-(\ep-16)^3\ep'$ & $C_2\times C_2$ \\
41A1$'$ $(E_{24})$& 1 & 0 & 0 & $-\ep'$ & 0 & $\ep^{\prime4}$ & $-(\ep'-16)^3\ep$ & $C_2\times C_2$ \\
41A2 $\phantom{'}(E_{26})$ & 1 & $(7-\sqrt{41})/2$ & $(7-\sqrt{41})/2$ & $6-\sqrt{41}$ & 0 & $-\ep'$ & $17^3\ep$ & $C_4$ \\
41A2$'$ $(E_{25})$ & 1 & $(7+\sqrt{41})/2$ & $(7+\sqrt{41})/2$ & $6+\sqrt{41}$ & 0 & $-\ep$ & $17^3\ep'$ & $C_4$ \\
41A3 $\phantom{'}(E_{28})$ & 1 & 0 & 0 & $4\ep$ & $\ep$ & $-\ep^5$ & $-(256\ep'+1)^3\ep$ & $C_2$ \\
41A3$'$ $(E_{27})$ & 1 & 0 & 0 & $4\ep'$ & $\ep'$ & $-\ep^{\prime5}$ & $-(256\ep+1)^3\ep'$ & $C_2$ \\
\hline
\end{tabular}

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\newpage
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
\hfil Code & $a_1$ & $a_2$ & $a_3$ & $a_4$ & $a_6$ & $\varDelta$ & $j$ & tors \\
\hline
65A1 $\phantom{'}(E_{29})$ & 1 & $-4\ep$ & 0 & $-\ep$ & 0 & $\ep^6$ & $257^3$ & $C_2\times C_2$  \\
65A2 $\phantom{'}(E_{31})$ & 1 & $2\ep$ & 0 & $\ep^2$ & 0 & $\ep^6$ & $17^3$ & $C_4\times C_2$  \\
65A3 $\phantom{'}(E_{38})$ & 1 & $-4\ep$ & 0 & $4\ep$ & $2\ep-\ep^3$ & $\ep^{\prime3}$ & $(256\ep^{\prime2}-\ep)^3$ & $C_2$  \\
65A3$'$ $(E_{37})$ & 1 & $-4\ep'$ & 0 & $4\ep'$ & $2\ep'-\ep^{\prime3}$ & $\ep^3$ & $(256\ep^2-\ep')^3$ & $C_2$  \\
65A4 $\phantom{'}(E_{35})$ & 1 & $-\omega$ & $\omega$ & $6+4\ep$ & $-(431+53\sqrt{65})/2$ & $-5^6\ep^3$ & $(8+\ep')^3$ & $C_4$  \\
65A4$'$ $(E_{36})$ & 1 & $-\omega'$ & $\omega'$ & $6+4\ep'$ & $-(431-53\sqrt{65})/2$ & $-5^6\ep^{\prime3}$ & $(8+\ep)^3$ & $C_4$  \\
\hline
\end{tabular}

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\bigskip\bigskip
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
\hfil Code & $a_1$ & $a_2$ & $a_3$ & $a_4$ & $a_6$ & $\varDelta$ & $j$ & tors \\
\hline
65B1 $\phantom{'}(E_{30})$ & 1 & $1-20\ep$ & 0 & $-25\ep$ & 0 & $(5\ep)^6$ & $257^3$ & $C_2\times C_2$ \\
65B2 $\phantom{'}(E_{32})$ &  1 & $1+10\ep$ & 0 & $25\ep^2$ & 0 & $(5\ep)^6$ & $17^3$ & $C_2\times C_2$ \\
65B3 $\phantom{'}(E_{40})$ & 1 & $1-20\ep$ & 0 & $100\ep$ & $-125(2\ep-\ep^3)$ & $-5^6\ep^9$ & $(256\ep^{\prime2}-\ep)^3$ & $C_2$ \\
65B3$'$ $(E_{39})$ & 1 & $1-20\ep'$ & 0 & $100\ep'$ & $-125(2\ep'-\ep^{\prime3})$ & $-5^6\ep^{\prime9}$ & $(256\ep^2-\ep')^3$ & $C_2$ \\
65B4 $\phantom{'}(E_{33})$ & 1 & $1+\omega$ & $\omega$ & $7+\omega$ & $\omega$ & $-\ep^3$ & $(8+\ep')^3$ & $C_2$ \\
65B4$'$ $(E_{34})$ & 1 & $1+\omega'$ & $\omega'$ & $7+\omega'$ & $\omega'$ & $-\ep^{\prime3}$ & $(8+\ep)^3$ & $C_2$ \\
\hline
\end{tabular}

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}

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\begin{flushright}
$B2C@n(B $B5.>O(B(Tak$@,$aaki KAGAWA) \\
$B")(B169--8555 $BAa0pEDBg3XM}9)3XIt>pJs3X2J(B \\
E-mail: k$@,$agawa@@mn.waseda.ac.jp
\end{flushright}

\end{document}
