THE IMAGE OF GALOIS REPRESENTATIONS ATTACHED TO ELLIPTIC CURVES WITH AN ISOGENY

We consider an elliptic curve E defined over Q which has an isogeny of prime degree p defined over Q. Assuming that E does not have complex multiplication and that p > 7, we show that the image of the Galois representation defined by the action of G(Q) on the p-adic Tate module is as large as possible, given the existence of such an isogeny. Under a certain additional assumption, we also prove that result for p = 7. For p = 5, we show that the image is as large as allowed by the isogenies of p-power degree defined over Q.