Explicit classification of isogeny graphs of rational elliptic curves

TL;DR

This paper explicitly classifies the isogeny graphs of rational elliptic curves with non-trivial isogenies over ields, using parameterized families and Fricke parameterizations, with implications for semistability after quadratic twists.

Contribution

It introduces 56 parameterized families of elliptic curves to classify isogeny graphs of rational elliptic curves with specific isogenies over fields.

Findings

Explicit classification of isogeny graphs for certain rational elliptic curves.

Construction of parameterized families with Fricke parameterizations.

Existence of quadratic twists that are semistable at almost all primes.

Abstract

Let $n>1$ be an integer such that $X_{0}\!\left( n\right) $ has genus $0$, and let $K$ be a field of characteristic $0$ or relatively prime to $6n$. In this article, we explicitly classify the isogeny graphs of all rational elliptic curves that admit a non-trivial isogeny over $\mathbb{Q}$. We achieve this by introducing $56$ parameterized families of elliptic curves $\mathcal{C}_{n,i}(t,d)$ defined over $K(t,d)$, which have the following two properties for a fixed $n$: the elliptic curves $\mathcal{C}_{n,i}(t,d)$ are isogenous over $K(t,d)$, and there are integers $k_{1}$ and $k_{2}$ such that the $j$-invariants of $\mathcal{C}_{n,k_{1}}(t,d)$ and $\mathcal{C}_{n,k_{2}}(t,d)$ are given by the Fricke parameterizations. As a consequence, we show that if $E$ is an elliptic curve over a number field $K$ with isogeny class degree divisible by $n\in\left\{4,6,9\right\} $, then there is a quadratic twist of $E$ that is semistable at all primes $\mathfrak{p}$ of $K$ such that $\mathfrak{p}\nmid n$.