On the family of elliptic curves y2 = x3 - m2x + p2

In this paper, we study the torsion subgroup and rank of elliptic curves for the subfamilies of Em, p : y2 = x3 - m2x + p2, where m is a positive integer and p is a prime. We prove that for any prime p, the torsion subgroup of Em, p(Q) is trivial for both the cases {m = 1, m = 0 (mod 3)} and {m = 1, m = 0 (mod 3), with gcd(m, p) = 1}. We also show that given any odd prime p and for any positive integer m with m = 0 (mod 3) and m = 2 (mod 32), the lower bound for the rank of Em, p(Q) is 2. Finally, we find curves of rank 9 in this family.