On the family of elliptic curves y²=x³-m²x + (pqr)²

In this article, we consider a family of elliptic curves defined by Eₘ: y²= x³ -m² x + (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 3, m ≡ 0 4 ~and~ m ≡ 2 2ᵏ where k ≥ 5, then the torsion subgroup of Eₘ(Q) is trivial and lower bound of the Q rank of this family of elliptic curves is $2$.