A classification of isogeny‐torsion graphs of Q ‐isogeny classes of elliptic curves

Abstract

Let E be a Q ‐isogeny class of elliptic curves defined over Q . The isogeny graph associated to E is a graph which has a vertex for each elliptic curve in the Q ‐isogeny class E , and an edge for each cyclic Q ‐isogeny of prime degree between elliptic curves in the isogeny class, with the degree recorded as a label of the edge. In this paper, we define an isogeny‐torsion graph to be an isogeny graph where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over Q of the corresponding elliptic curve. Then, the main result of the paper is a classification of all the possible isogeny‐torsion graphs that occur for Q ‐isogeny classes of elliptic curves defined over the rationals.