Computing Isogenies at Singular Points of the Modular Polynomial

William E. Mahaney; Travis Morrison

paper

Computing Isogenies at Singular Points of the Modular Polynomial

arXiv:2402.02038

Abstract

In this paper we present a method which, given a singular point $(j_{1}, j_{2})$ on $Y_{0} (ℓ)$ with $j_{1}, j_{2} \neq = 0, 1728$ and an elliptic curve $E$ with $j$ -invariant $j_{1}$ , returns an elliptic curve $E$ with $j$ -invariant $j_{2}$ that admits a normalized $ℓ$ -isogeny $ϕ : E \to E$ . The isogeny $ϕ$ can then be efficiently computed from $E$ and $E$ using an algorithm of Bostan, Morain, Salvy, and Schost.