Radical characterizations of elliptic curves

Let K be a number field, and let E be an elliptic curve over K. A famous result by Faltings of 1983 can be reformulated for elliptic curves as follows: if S is a set of primes of good reduction for E having density one, then the K-isogeny class of E is determined by the function which maps a prime in S to the size of the group of points over the residue field. In this paper, we prove that it suffices to look at the radical of the size.