Let Cm:y2 = x3 − m2x + p2q2 be a family of elliptic curves over ℚ, where m is a positive integer and p, q are distinct odd primes. We study the torsion part and the rank of Cm(ℚ). More specifically, we prove that the torsion subgroup of Cm(ℚ) is trivial and the ℚ-rank of this family is at least 2, whenever m ≢ 0 (mod 3), m ≢ 0 (mod 4) and m ≡ 2 (mod 64) with neither p nor q dividing m.