Average normalisations of elliptic curves

Ciet, Quisquater, and Sica have recently shown that every elliptic curve E over a finite field 𝔽p is isomorphic to a curve y2 = x3 + ax + b with a and b of size O (p¾). In this paper, we show that almost all elliptic curves satisfy the stronger bound O (p⅔). The problem is motivated by cryptographic considerations.