  Watkins' conjecture asserts that for a rational elliptic curve $E$ the degree
of the modular parametrization is divisible by $2^r$, where $r$ is the rank of
$E$. In this paper we prove that if the modular degree is odd then $E$ has rank
$0$. Moreover, we prove that the conjecture holds for all rank two rational
elliptic curves of prime conductor and positive discriminant.
