\magnification 1200
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\nopagenumbers
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\centerline{{\bf Multiplicities}}
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\noindent
Define $\mu(N)$ to be the number of elliptic curve factors of $J_0(N)$.

\noindent
Using results of MESTRE (rank bound in terms of conductor, under GRH) and 
SILVERMAN (number of integral points bounded by function of rank), BRUMER
observed that
$$
\mu(N)= {\cal O}\left(N^{c\over\log\log N}\right)
$$
for some constant~$c$.
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\noindent
If $\mu(N)$ is unbounded for $N$ prime, $N\to\infty$, then the rank of the curve
$Y^2=X^3\pm N$ is unbounded, as is the 3-rank of the class group of ${\bf
Q}(\sqrt{N})$.
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\noindent
For conductor $N=61,263,451$ there are 13~curves of this conductor.
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\centerline{\bf PRELIMINARY DISTRIBUTION}
$$
\vbox{
\settabs 2\columns
\+$\mu$& Number of Curves\cr
\+ 1 & $\leq 268020$\cr
\+2 & \hfill$\geq 15004$\cr
\+3 & \hfill$\geq 3267$\cr
\+ 4 & \hfill$\geq 487$\cr
\+5 & 159\cr
\+6 & 49 \cr
\+ 7 & 26\cr
\+8 & 7\cr
\+9 & 7\cr
\+10 & 4 \cr
\+ 13 & 1\cr
}
$$

 \bye