Computing the height of volcanoes of ℓ-isogenies of elliptic curves over finite fields

The structure of the volcano of ℓ-isogenies, ℓ-prime, of elliptic curves over finite fields has been extensively studied over recent years. Previous works present some results and algorithms concerning the height of such volcanoes in the case of isogenies whose kernels are generated by a rational point. The main goal of this paper is to extend such works to the case of ℓ-isogenies whose kernels are defined by a rational subgroup. In particular, the height of such volcanoes is completely characterized and can be computationally obtained.