Abstract. Let E be an elliptic curve defined over the rational numbers. For a prime p of good reduction for E, let E_p denote the reduction of E modulo p. There are many conjectures which give precise asymptotics for functions which count the number of primes p up to x for which E_p has some desired property (e.g. its number of points is prime, or is equal to p+1-r for a fixed integer r). In this talk I will use a square-free sieve of Hooley to study the constants appearing in these asymptotic formulas.