Sato-Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height

We obtain asymptotic formulae for the number of primes p ≤ 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 such that |a| ? A and |b| ? B, where A and B are small relative to x. More precisely, we investigate behavior with respect to the Sato-Tate conjecture, cyclicity, and divisibility of the number of points by a fixed integer m.