Isomorphism classes of elliptic curves over a finite field in some thin families

Foraprime p and a given square box, B, we consider all elliptic curves E _r,s: Y ² = X ³ +rX +s defined over a field F _pp of p elements with coefficients (r, s) ∈ B. We obtain a nontrivial upper bound for the number of such curves which are isomorphic to a given one over F _p, in terms of the size of B. We also give an optimal lower bound on the number of distinct isomorphic classes represented.