In this paper we consider the problem of classifying quadruples (K,E,m₁,m₂) where K is a number field, E is an elliptic curve defined over K and (m₁,m₂) is a pair of relatively prime positive integers for which the intersection K(E[m₁]) ∩ K(E[m₂]) is a non-abelian extension of K. There is an infinite set S of modular curves whose K-rational points capture all elliptic curves over K without complex multiplication that have this property. Our main theorem explicitly describes the subset S₀ ⊆ S consisting of those modular curves having genus zero. The subset S₀ turns out to consist of four modular curves, each isomorphic to P¹ over its field of definition. In the case K = Q, this has applications to the problem of determining when the Galois representation on the torsion of E is as large as possible modulo a prescribed obstruction; we illustrate this application with a specific example.

The authors would like to thank David Zureick-Brown for insightful conversations and also Jackson Morrow and Harris Daniels, as well as Andrew Sutherland, for helpful comments on an earlier version of the paper. Finally, we thank the anonymous referee for a careful reading of the manuscript and many helpful suggestions for improvement.