Rank of the family of elliptic curves $y^2 = x^3- 5px$

Abstract

This article considers the family of elliptic curves given by $E_{p}: y^2=x^3-5px$ and certain conditions on an odd prime $p$. More specifically, we have shown that if $p \equiv 7, 23 \pmod {40}$, then the rank of $E_{p}$ is zero for both $ \mathbb{Q} $ and $ \mathbb{Q}(i) $. Furthermore, if the prime $ p $ is of the form $ 40k_1 + 3 $ or $ 40k_2 + 27$, where $k_1, k_2 \in \mathbb{Z}$ such that $(5k_1+1)$ or $(5k_2 +4)$ are perfect squares, then the given family of elliptic curves has rank one over $\mathbb{Q}$ and rank two over $\mathbb{Q}(i)$. Moreover, if the prime $ p $ is of the form $ 40k_3 + 11 $ or $ 40k_4 + 19$ where $k_3 ~\text{and}~ k_4 \in \mathbb{Z}$ such that $(160k_3+49)$ or $(160k_4 + 81) $ are perfect squares, then the given family of elliptic curves has rank at least one over $\mathbb{Q}$ and rank at least two over $\mathbb{Q}(i)$.