A Family of Elliptic Curves of Rank One

In this article, we study the family of elliptic curves given by E_p: y^2=x^3-5px for an odd prime p 5 and try to find the conditions under which E_p has rank one. Specifically, we have shown that if p ≡ 3, 27 40 such that 2(p+5) is a perfect square or if p ≡ 11, 19 40 such that (5+4p) is a perfect square, then the Mordell-Weil rank of E_p is exactly one. Finally, assuming the parity conjecture, we have proved that if p ≡ 3, 11, 19, 27 40 , then the rank of E_p is always one.