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\title{The unusual Hecke algebra of $X_0(389)$ }
\author{William A.~Stein\footnote{UC Berkeley, 
Department of Mathematics, Berkeley, CA  94720, USA.}}
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\begin{displaymath}J \sim A_1\times A_2 \times A_3 \times A_6 \times A_{20},\end{displaymath}%
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$\dim A_d = d$%
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$J_0(N)^{-}\sim A_{1} \times A_{20}.$%
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\begin{proposition}The Mordell-Weil ranks of the factors are as follows:
\begin{center}
\begin{tabular}{|l|lllll|}\hline
Factor& $A_1$\space & $A_2$\space & $A_3$\space & $A_6$\space & $A_{20}$\\\hline
Rank & 2 & ? & ? & ? & 0\\\hline
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\end{proposition}%
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\begin{proof}% latex2html id marker 50
The elliptic curve $A_1$\space is $389A$\space in Cremona's tables, which is
the elliptic curve of smallest conductor having rank $2$ . 
\par For $A_{20}$\space we directly compute whether or not the $L$ -function
vanishes using modular symbols, by taking an inner product with
the winding element $e_w=-\{0,\infty\}$ .  We find that the $L$ -function
does not vanish.  By Kolyvagin-Logachev, it follows that $A_{20}$\space has
Mordell-Weil rank $0$ . 
\par I don't know the ranks of the other three factors, but the sign of 
the functional equation is odd so the rank is expected to be odd. 
\end{proof}%
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$A_1(\mathbf{Q})=\mathbf{Z}\oplus \mathbf{Z}$%
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\begin{eqnarray*}T_2&=&(x + 2)({x^{2} - {2}})({x^{3} - {4}x - {2}})(x^{6} + {3}x^{5} - {2}x^{4} - {8}x^{3} + {2}x^{2} + {4}x - 1)\\
 &&(x^{20} - {3}x^{19} - {29}x^{18} + {91}x^{17} + {338}x^{16} - {1130}x^{15} - {2023}x^{14}+ {7432}x^{13}+ {6558}x^{12}\\
 &&- {28021}x^{11} - {10909}x^{10} + {61267}x^{9} + {6954}x^{8}- {74752}x^{7} + {1407}x^{6} + {46330}x^{5}\\
 && - {1087}x^{4} - {12558}x^{3} - {942}x^{2} + {960}x + {148})\\
T_3&=&(x + 2)(x^2 + 4x + 2)(x^3 -4x + 2)(x^6 + 5x^5 + 4x^4 -13x^3 -21x^2 -6x + 1)\\
 &&(x^{20} -11x^{19} + 19x^{18} + 204x^{17} -845x^{16} -781x^{15} + 8883x^{14} -6177x^{13} -40916x^{12}\\
 &&+ 63058x^{11} + 85034x^{10} -215618x^9 -46920x^8 + 342529x^7 -84612x^6 -241030x^5 \\
 &&+ 112365x^4 + 51018x^3 -28526x^2 + 3560x -100)
\end{eqnarray*}%
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$S_2(\Gamma_0(389))$%
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$\alpha^2-2=0$%
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$\mathbf{T}\subset\mbox{\rm End}(S_2(\Gamma_0(N)))$%
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$\mathbf{T}\otimes\mathbf{Q}$%
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$K_1\times\cdots\times K_n$%
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$\tilde{\mathbf{T}}$%
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$\O_i$%
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\begin{displaymath}2^{53}\cdot{}3^4\cdot{}5^6\cdot{}31^2\cdot{}37\cdot{}389
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$T_1,\ldots, T_{65}$%
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\begin{eqnarray*}K_1&=&\mathbf{Q}, \\
K_2&=&\mathbf{Q}(\sqrt{2})\\
K_3&=&\mathbf{Q}(\beta) , \quad \beta^3-4\beta-2=0,\\
K_6&=&\mathbf{Q}(\gamma), 
      \quad \gamma^6+3\gamma^5-2\gamma^4-8\gamma^3+2\gamma^2+4\gamma-1=0,\\
K_{20}&=&\mathbf{Q}(\delta), \quad \delta^{20}-3\delta^{19}-29\delta^{18}+91\delta^{17}+338\delta^{16}-1130\delta^{15}-2023\delta^{14}+7432\delta^{13}\\
    &&\qquad +6558\delta^{12}-28021\delta^{11}-10909\delta^{10}+61267\delta^9 +6954\delta^8-74752\delta^7\\
    &&\qquad +1407\delta^6+46330\delta^5
-1087\delta^4-12558\delta^3-942\delta^2+960\delta+148=0.
\end{eqnarray*}%
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$\quad 2^2\cdot 37$%
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$\quad 5^3\cdot 3881$%
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$\quad 2^{14}\cdot 5\cdot 389 \cdot 215517113148241\cdot 477439237737571441$%
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