On a conjecture of Deines

TL;DR

This paper proves Deines' conjecture by constructing an explicit infinite family of semi-stable, non-isogenous elliptic curves over Q that share the same minimal discriminant and conductor.

Contribution

It provides the first explicit infinite family of discriminant twins, confirming the conjecture for semi-stable non-isogenous elliptic curves.

Findings

Constructed an explicit infinite family of discriminant twins.

Proved the existence of infinitely many semi-stable non-isogenous discriminant twins.

Confirmed Deines' conjecture in the semi-stable case.

Abstract

Two elliptic curves defined over $\mathbb{Q}$ are called discriminant twins if they have the same minimal discriminant and the same conductor. Deines, in 2014, conjectured that there exist infinitely many semi-stable non-isogenous discriminant twins. In this article we present an explicit infinite family of semi-stable non-isogenous discriminant twins, providing a proof for Deines' conjecture.