Abstract
Let n > 1 be an integer such that X-0 (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 C-n,C- i(t,d) defined over K(t, d), which have the following two properties for a fixed n: the elliptic curves C-n,C- i(t,d) are isogenous over K(t, d), and there are integers k(1) and k(2) such that the j-invariants of C-n,C- k1 (t, d) and C-n,C- 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 is an element of {4, 6, 9}, then there is a quadratic twist of E that is semistable at all primes p of K such that p inverted iota n.