On the family of elliptic curves $$varvec{y^2=x^3-m^2x+p^2}$$

In this paper, we study the torsion subgroup and rank of elliptic curves for the subfamilies of (E_{m,p} : y^2=x^3-m^2x+p^2), where m is a positive integer and p is a prime. We prove that for any prime p, the torsion subgroup of (E_{m,p}(mathbb {Q})) is trivial for both the cases {(mge 1), (mnot equiv 0pmod 3)} and {(mge 1), (m equiv 0 pmod 3), with (gcd(m,p)=1)}. We also show that given any odd prime p and for any positive integer m with (mnot equiv 0pmod 3) and (mequiv 2pmod {32}), the lower bound for the rank of (E_{m,p}(mathbb {Q})) is 2. Finally, we find curves of rank 9 in this family.