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

In this article, we consider a family of elliptic curves defined by E₌: 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 E₌ (Q). More specifically, we proved that if m 0 3, m 0 4 ~and~ m 2 2^{k} where k 5, then the torsion subgroup of E₌ (Q) is trivial and lower bound of the Q rank of this family of elliptic curves is 2.