2-Adic Galois Images of Non-CM Isogeny-Torsion Graphs Defined over ℚ

Abstract

Let E be a Q -isogeny class of elliptic curves defined over Q without CM. The isogeny graph associated to E is a graph which has a vertex for each elliptic curve in E and an edge for each Q -isogeny of prime degree that maps one elliptic curve in E to another elliptic curve in E , with the degree recorded as a label of the edge. An isogeny-torsion graph is 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 statement of the article is a classification of the 2-adic Galois image that occurs at each vertex of all isogeny-torsion graphs consisting of elliptic curves defined over Q without CM.