Counting elliptic curves over the rationals with a 7-isogeny

TL;DR

This paper counts elliptic curves over the rationals with a 7-isogeny, considering both isomorphism classes over the rationals and algebraic closures, using height as a measure.

Contribution

It provides a quantitative count of elliptic curves with a 7-isogeny over the rationals, a specific case in the classification of elliptic curves.

Findings

Quantifies the number of such elliptic curves up to a certain height.

Distinguishes counts over the rationals and algebraic closures.

Advances understanding of elliptic curves with specific isogenies.

Abstract

We count by height the number of elliptic curves over the rationals, both up to isomorphism over the rationals and over an algebraic closure thereof, that admit a cyclic isogeny of degree $7$.