Elliptic curves of odd modular degree

The modular degree m _E of an elliptic curve E/Q is the minimal degree of any surjective morphism X ₀(N) → E, where N is the conductor of E. We give a necessary set of criteria for m _E to be odd. In the case when N is prime our results imply a conjecture of Mark Watkins. As a technical tool, we prove a certain multiplicity one result at the prime p = 2, which may be of independent interest. Supported in part by the American Institute of Mathematics. Supported in part by NSF grant DMS-0401545.