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 $p\equiv 7,23(\mathrm{mod}40)$, 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 $\mathbb{Q}(i)$.

There are some typoes in the previous version and conditions of p in torsor lemmas were not mentioned properly. I have updated those in this version