A well-known folk conjecture predicts that the average rank of an elliptic curve over Q is 1/2. Given an irreducible Artin representation r of Q, one can ask for the average multiplicity of r in the Mordell-Weil group over Qbar of an elliptic curve over Q. If r is the trivial one-dimensional representation then one recovers the average rank. To illustrate some issues that arise if one wants to formulate an ""average multiplicity"" conjecture, we shall look at an example where r has dimension 4 and Schur index 1 and factors though a Galois group over Q of order 32.