The image of Galois representations attached to elliptic curves with an isogeny

We consider an elliptic curve $E$ defined over ${\bf Q}$ which has an isogeny of prime degree $p$ defined over ${\bf Q}$. 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 $G_{{\bf Q}}$ 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 ${\bf Q}$.