  We obtain asymptotic formulae for the number of primes $p\le x$ for which the
reduction modulo $p$ of the elliptic curve $$ \E_{a,b} : Y^2 = X^3 + aX + b $$
satisfies certain ``natural'' properties, on average over integers $a$ and $b$
with $|a|\le A$ and $|b| \le B$, where $A$ and $B$ are small relative to $x$.
Specifically, we investigate behavior with respect to the Sato--Tate
conjecture, cyclicity, and divisibility of the number of points by a fixed
integer $m$.
