# LaTeX2HTML 98.1p1 release (March 2nd, 1998)
# Associate images original text with physical files.


$key = q/{inline}J_0(N)^-simA_1timesA_20.{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img3.gif"
 ALT="$J_0(N)^{-}\sim A_{1} \times A_{20}.$">|; 

$key = q/{displaymath}[tildemathbfT:mathbfT]=2^17cdot3^2cdot5cdot31.{displaymath}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="36"
 SRC="img36.gif"
 ALT="\begin{displaymath}[\tilde{\mathbf{T}}:\mathbf{T}]=2^{17}\cdot 3^2\cdot 5\cdot 31.\end{displaymath}">|; 

$key = q/{eqnarray*}T_2&=&(x+2)(x^2-2)(x^3-4x-2)(x^6+3x^5-2x^4-8x^3+2x^2+4x-1)&&(x^20-3x^612x^6-241030x^5&&+112365x^4+51018x^3-28526x^2+3560x-100){eqnarray*}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="756" HEIGHT="240"
 SRC="img7.gif"
 ALT="\begin{eqnarray*}T_2&=&(x + 2)({x^{2} - {2}})({x^{3} - {4}x - {2}})(x^{6} + {3}x...
...-241030x^5 \\\\
&&+ 112365x^4 + 51018x^3 -28526x^2 + 3560x -100)
\end{eqnarray*}">|; 

$key = q/{inline}K_1timescdotstimesK_n{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img19.gif"
 ALT="$K_1\times\cdots\times K_n$">|; 

$key = q/{inline}quad2^14cdot5cdot389cdot215517113148241cdot477439237737571441{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="468" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="img33.gif"
 ALT="$\quad 2^{14}\cdot 5\cdot 389 \cdot 215517113148241\cdot 477439237737571441$">|; 

$key = q/{proposition}TheMordell-Weilranksofthefactorsareasfollows:centertabular|l|lllll|e&inlineA_20inlinehlineRank&2&?&?&?&0hlinetabularcenter{proposition}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="447" HEIGHT="82" ALIGN="BOTTOM" BORDER="0"
 SRC="img4.gif"
 ALT="\begin{proposition}The Mordell-Weil ranks of the factors are as follows:
\begin{...
...ine
Rank & 2 & ? & ? & ? & 0\\\\ \hline
\end{tabular}\end{center}\end{proposition}">|; 

$key = q/{inline}mboxrmSpec(mathbfT){inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="img25.gif"
 ALT="$\mbox{\rm Spec}(\mathbf{T})$">|; 

$key = q/{inline}gamma^6+3gamma^5-2gamma^4-8gamma^3{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="img14.gif"
 ALT="$\gamma^6+3\gamma^5-2\gamma^4-8\gamma^3$">|; 

$key = q/{inline}dimA_d=d{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img2.gif"
 ALT="$\dim A_d = d$">|; 

$key = q/{inline}alpha^2-2=0{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="img10.gif"
 ALT="$\alpha^2-2=0$">|; 

$key = q/{displaymath}2^19cdot5^4cdot37cdot389cdot3881cdot215517113148241cdot477439237737571441.{displaymath}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="546" HEIGHT="31"
 SRC="img34.gif"
 ALT="\begin{displaymath}2^{19}\cdot 5^4\cdot 37\cdot 389\cdot 3881\cdot 215517113148241\cdot
477439237737571441.\end{displaymath}">|; 

$key = q/{inline}q+gammaq^2+(gamma^5+3gamma^4-2gamma^3-8gamma^2+gamma+2)q^3+cdots{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="419" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="img15.gif"
 ALT="${q + {\gamma}q^{2}+({\gamma^{5} + {3}\gamma^{4} - {2}\gamma^{3} - {8}\gamma^{2} + \gamma + {2}})q^{3}+\cdots}$">|; 

$key = q/{inline}q+betaq^2-betaq^3+(beta^2-2)q^4+(-beta^2+1)q^5-beta^2q^6+cdots{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="467" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="img13.gif"
 ALT="${q + {\beta}q^{2} - {\beta}q^{3}+({\beta^{2} - {2}})q^{4}+({-\beta^{2} + 1})q^{5} - {\beta^{2}}q^{6} +\cdots}$">|; 

$key = q/{inline}mathbfTotimesmathbfQ=K_1timesK_2timesK_3timesK_6timesK_20,{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="321" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img29.gif"
 ALT="$\mathbf{T}\otimes\mathbf{Q}= K_1\times K_2\times K_3\times K_6\times K_{20},$">|; 

$key = q/{inline}A_1(mathbfQ)=mathbfZoplusmathbfZ{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="img6.gif"
 ALT="$A_1(\mathbf{Q})=\mathbf{Z}\oplus \mathbf{Z}$">|; 

$key = q/{inline}[tildemathbfT:mathbfT]cdotprod_i=1^nmboxrmdisc(K_i){inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="img23.gif"
 ALT="$[\tilde{\mathbf{T}}:\mathbf{T}]\cdot \prod_{i=1}^n\mbox{\rm disc}(K_i)$">|; 

$key = q/{inline}q-2q^2-2q^3+2q^4-3q^5+4q^6-5q^7+q^9+6q^10+cdots{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="469" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="img9.gif"
 ALT="${q - {2}q^{2} - {2}q^{3} + {2}q^{4} - {3}q^{5} + {4}q^{6} - {5}q^{7} + q^{9} + {6}q^{10} + \cdots }$">|; 

$key = q/{inline}quad5^3cdot3881{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="img32.gif"
 ALT="$\quad 5^3\cdot 3881$">|; 

$key = q/{displaymath}2^53cdot3^4cdot5^6cdot31^2cdot37cdot389cdot3881cdot215517113148241cdot477439237737571441.{displaymath}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="582" HEIGHT="31"
 SRC="img27.gif"
 ALT="\begin{displaymath}2^{53}\cdot{}3^4\cdot{}5^6\cdot{}31^2\cdot{}37\cdot{}389
\cdot{}3881\cdot{}215517113148241\cdot{}477439237737571441.\end{displaymath}">|; 

$key = q/{inline}[tildemathbfT:mathbfT]{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="img26.gif"
 ALT="$[\tilde{\mathbf{T}}:\mathbf{T}]$">|; 

$key = q/{eqnarray*}K_1&=&mathbfQ,K_2&=&mathbfQ(sqrt2)K_3&=&mathbfQ(beta),quadbeta^3-4betlta^5-1087delta^4-12558delta^3-942delta^2+960delta+148=0.{eqnarray*}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="709" HEIGHT="207"
 SRC="img30.gif"
 ALT="\begin{eqnarray*}K_1&=&\mathbf{Q}, \\\\
K_2&=&\mathbf{Q}(\sqrt{2})\\\\
K_3&=&\math...
...delta^5
-1087\delta^4-12558\delta^3-942\delta^2+960\delta+148=0.
\end{eqnarray*}">|; 

$key = q/{inline}2^34cdot3^4cdot5^2cdot31^2{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="img35.gif"
 ALT="$2^{34}\cdot 3^4\cdot 5^2\cdot 31^2$">|; 

$key = q/{inline}prod_i=1^nmboxrmdisc(K_i){inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="img24.gif"
 ALT="$\prod_{i=1}^n\mbox{\rm disc}(K_i)$">|; 

$key = q/{inline}beta^3-4beta-2=0{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="img12.gif"
 ALT="$\beta^3 - 4\beta - 2=0$">|; 

$key = q/{inline}O_i{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img22.gif"
 ALT="$\O_i$">|; 

$key = q/{displaymath}JsimA_1timesA_2timesA_3timesA_6timesA_20,{displaymath}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="35"
 SRC="img1.gif"
 ALT="\begin{displaymath}J \sim A_1\times A_2 \times A_3 \times A_6 \times A_{20},\end{displaymath}">|; 

$key = q/{inline}tildemathbfT{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
 SRC="img20.gif"
 ALT="$\tilde{\mathbf{T}}$">|; 

$key = q/{inline}mathbfTotimesmathbfQ{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img18.gif"
 ALT="$\mathbf{T}\otimes\mathbf{Q}$">|; 

$key = q/{inline}S_2(Gamma_0(389)){inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.gif"
 ALT="$S_2(\Gamma_0(389))$">|; 

$key = q/{inline}+2gamma^2+4gamma-1=0{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="img16.gif"
 ALT="$+2\gamma^2+4\gamma-1=0$">|; 

$key = q/{inline}mathbfTsubsetmboxrmEnd(S_2(Gamma_0(N))){inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="img17.gif"
 ALT="$\mathbf{T}\subset\mbox{\rm End}(S_2(\Gamma_0(N)))$">|; 

$key = q/{inline}T_1,ldots,T_65{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img28.gif"
 ALT="$T_1,\ldots, T_{65}$">|; 

$key = q/{inline}quad2^2cdot37{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="img31.gif"
 ALT="$\quad 2^2\cdot 37$">|; 

$key = q/{inline}tildemathbfT=prodO_i{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="img21.gif"
 ALT="$\tilde{\mathbf{T}}=\prod \O_i$">|; 

$key = q/{inline}q+alphaq^2+(alpha-2)q^3-q^5+(-2alpha+2)q^6+cdots{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="389" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="img11.gif"
 ALT="${q + {\alpha}q^{2}+({\alpha- {2}})q^{3} - q^{5}+({{-2}\alpha+ {2}})q^{6}+ \cdots}$">|; 

$key = q/{proof}%latex2htmlidmarker50TheellipticcurveinlineA_1inlinespaceisinline389Ainlifthefunctionalequationisoddsotherankisexpectedtobeodd.{proof}MSF=2.5;TAGS=R;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="584" HEIGHT="211" ALIGN="BOTTOM" BORDER="0"
 SRC="img5.gif"
 ALT="\begin{proof}% latex2html id marker 50
The elliptic curve $A_1$\space is $389A$\...
...
the functional equation is odd so the rank is expected to be odd.
\end{proof}">|; 

1;

