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Let $\{E_{(p,q)}\}$ be a family of elliptic curves over a rational field suchthat we have $E_{(p,q)} : y^2 = x^3 - p^2x + q^2$, where $p$ and $q$ are primenumbers 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 independentpoints. 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.

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