Discriminants of complex multiplication fields of elliptic curves over finite fields

We show that, for most of the elliptic curves E over a prime finite field double script F sign_p of p elements, the discriminant D(E) of the quadratic number field containing the endomorphism ring of E over double script F sign_p is sufficiently large. We also obtain an asymptotic formula for the number of distinct quadratic number fields generated by the endomorphism rings of all elliptic curves over double script F sign_p.