Explicit classification of isogeny graphs of rational elliptic curves

Abstract

Let [Formula: see text] be an integer such that [Formula: see text] has genus [Formula: see text], and let [Formula: see text] be a field of characteristic [Formula: see text] or relatively prime to [Formula: see text]. In this paper, we explicitly classify the isogeny graphs of all rational elliptic curves that admit a nontrivial isogeny over [Formula: see text]. We achieve this by introducing [Formula: see text] parameterized families of elliptic curves [Formula: see text] defined over [Formula: see text], which have the following two properties for a fixed [Formula: see text]: the elliptic curves [Formula: see text] are isogenous over [Formula: see text], and there are integers [Formula: see text] and [Formula: see text] such that the [Formula: see text]-invariants of [Formula: see text] and [Formula: see text] are given by the Fricke parameterizations. As a consequence, we show that if [Formula: see text] is an elliptic curve over a number field [Formula: see text] with isogeny class degree divisible by [Formula: see text], then there is a quadratic twist of [Formula: see text] that is semistable at all primes [Formula: see text] of [Formula: see text] such that [Formula: see text].