On the elliptic curve $ y^{2} = x (x^2 + p)$ over some certain imaginary quadratic fields

TL;DR

This paper studies the properties of a specific elliptic curve over certain imaginary quadratic fields, focusing on fields where the discriminant conditions and coprimality influence the curve's behavior.

Contribution

It investigates the elliptic curve $ y^{2} = x (x^2 + p)$ over imaginary quadratic fields with particular discriminant and coprimality conditions, extending understanding of elliptic curves in these settings.

Findings

Analysis of the elliptic curve over $Q( oot{-q})$ with $q mod 8=3$

Conditions under which the curve exhibits specific properties

Insights into the structure of the Mordell-Weil group over these fields

Abstract

In this paper, we will talk about the titled elliptic curve defined over imaginary quadratic fields such as $\mathbb{Q}(\sqrt{-q})$, where $q$ is congruent to 3 modulo 8 and $(p,q)=1$.