  Let $E$ be an elliptic curve without CM that is defined over a number field
$K$. For all but finitely many nonarchimedean places $v$ of $K$ there is the
reduction $E(v)$ of $E$ at $v$ that is an elliptic curve over the residue field
$k(v)$ at $v$. The set of $v$'s with ordinary $E(v)$ has density 1 (Serre). For
such $v$ the endomorphism ring $End(E(v))$ of $E(v)$ is an order in an
imaginary quadratic field.
  We prove that for any pair of relatively prime positive integers $N$ and $M$
there are infinitely many nonarchimedean places $v$ of $K$ such that the
discriminant $\Delta(v)$ of $End(E(v))$ is divisible by $N$ and the ratio
$\Delta(v)/N$ is relatively prime to $NM$. We also discuss similar questions
for reductions of abelian varieties.
  The subject of this paper was inspired by an exercise in Serre's "Abelian
$\ell$-adic representations and elliptic curves" and questions of Mihran
Papikian and Alina Cojocaru.
