Distorting the volcano

Volcanoes of ℓ-isogenies of elliptic curves are a special case of graphs with a cycle called crater. In this paper, given an elliptic curve E of a volcano of ℓ-isogenies, we present a condition over an endomorphism φ of E in order to determine which ℓ-isogenies of E are non-descending. The endomorphism φ is defined as the crater cycle of an m-volcano where E is located, with m≠ℓ. The condition is feasible when φ is a distortion map for a subgroup of order ℓ of E. We also provide some relationships among the crater sizes of volcanoes of m-isogenies whose elliptic curves belong to a volcano of ℓ-isogenies.