Let E/Q be an elliptic curve, G a finite group and V a fixed finite dimensional rational representation of G. As we run over finite Galois extensions F/Q with Galois group G and E(F) \otimes Q isomorphic to V, how does the Z[G]-module structure of E(F) vary from a statistical point of view? I will report on joint work with Alex Bartel in which we propose a heuristic giving a conjectural answer to an instance of this question, and discuss work in progress proving parts of this. In the process I will explain a close analogy between these heuristics and Stevenhagen’s conjecture concerning the solubility of the negative Pell equation.