Explicit classification of isogeny graphs of rational elliptic curves

Let $n>1$ be an integer such that $X_0\text{\hspace{0.17em}⁣}\left(n\right)$ 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 $n\in \{4,6,9\}$, then there is a quadratic twist of $E$ that is semistable at all primes $\mathfrak{p}$ of $K$ such that $\mathfrak{p}\nmid n$.