On the squareness of the discriminant of elliptic curves with an isogeny

TL;DR

This paper classifies the integers N for which elliptic curves over Q with square discriminant admit N-isogenies, and parametrizes the related j-invariants, advancing understanding of elliptic curve isogenies with square discriminants.

Contribution

It provides a complete classification and explicit parametrization of N for elliptic curves over Q with square discriminant related to N-isogenies.

Findings

Classified N values for N-isogenies of elliptic curves with square discriminant.

Explicit parametrization of j-invariants for these elliptic curves.

Determined N values for pairs of such elliptic curves to be N-isogenous.

Abstract

We establish a classification of the values of \( N \) for which an elliptic curve defined over \( \mathbb{Q} \) with square discriminant admits an \( N \)-isogeny. Furthermore, we determine the values of \( N \) for which two elliptic curves defined over \( \mathbb{Q} \), both possessing square discriminants, are \( N \)-isogenous. In both cases, we explicitly parametrize the corresponding \( j \)-invariants of the elliptic curves associated with these problems.