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. 
Chris Hall, Antonella Perucca
math.NT