Generators for the elliptic curve $E_{(p,q)} : y^2 = x^3 - p^2x + q^2$

Abstract: Let ${E_{(p,q)}}$ be a family of elliptic curves over a rational field such that we have $E_{(p,q)} : y^2 = x^3 - p^2x + q^2$, where $p$ and $q$ are prime numbers greater than five. Earlier work showed that the elliptic curve $E_{(p,q)}$ had ranked at least two for all $p, q > 5$ and two independent points. This paper shows that two points that can be extended to a basis for $E_{(p,q)}$ under conditions are confident that we will fully recover.