Explicit classification of isogeny graphs of rational elliptic curves

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