Explicit classification of isogeny graphs of rational elliptic curves

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 $\mathfrak{p}\nmid n$.