Explicit classification of isogeny graphs of rational elliptic curves

Abstract

Let n > 1 be an integer such that X0(n) has genus 0, and let K be a field of characteristic 0 or relatively prime to 6n. In this paper, we explicitly classify the isogeny graphs of all rational elliptic curves that admit a nontrivial isogeny over ℚ. We achieve this by introducing 56 parameterized families of elliptic curves n,i(t,d) defined over K(t,d), which have the following two properties for a fixed n: the elliptic curves n,i(t,d) are isogenous over K(t,d), and there are integers k1 and k2 such that the j-invariants of n,k1(t,d) and n,k2(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 {4, 6, 9}, then there is a quadratic twist of E that is semistable at all primes of K such that n. ;copy; 2023 World Scientific Publishing Company.