Isogeny volcanoes of elliptic curves and sylow subgroups

Abstract

Given an ordinary elliptic curve over a finite field located in the floor of its volcano of ℓ-isogenies, we present an efficient procedure to take an ascending path from the floor to the level of stability and back to the floor. As an application for regular volcanoes, we give an algorithm to compute all the vertices of their craters. In order to do this, we make use of the structure and generators of the ℓ-Sylow subgroups of the elliptic curves in the volcanoes.