Compact elliptic curve representations

Let y² = x³ + ax + b be an elliptic curve over F;_p, p being a prime number greater than 3, and consider a; b ∈ [1, p]. In this paper, we study elliptic curve isomorphisms, with a view towards reduction in the size of elliptic curves coefficients. We first consider reducing the ratio a/b. We then apply these considerations to determine the number of elliptic curve isomorphism classes. Later we work on both coefficients. We introduce the number M.p/as the lower bound of all M ∈ ℕ such that each isomorphism class has a representative with max (a,b) < M. Using results from the theory of uniform distributions, we prove upper and lower bounds of the form c₁p1/2 < M(p) < c₂p ^3/4 with explicit constants c₁, c₂ > 0.