For a prime 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 p 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.