In this article, we consider a family of elliptic curves defined by Em: y2 = x3 − m2x + (pqr)2 where m is a positive integer and p, q, and r are distinct odd primes and study the torsion as well the rank of Em(Q). More specifically, we proved that if m ≢ 0 (mod 3), m ≡ 2 (mod 2k) where k ≥ 5, and none of the prime numbers p, q and r divide m, then the torsion subgroup of Em(Q) is trivial and a lower bound of the Q rank of this family of elliptic curves is 2.