  Watkins conjectured that for an elliptic curve $E$ over $\mathbb{Q}$ of
Mordell-Weil rank $r$, the modular degree of $E$ is divisible by $2^r$. If $E$
has non-trivial rational $2$-torsion, we prove the conjecture for all the
quadratic twists of $E$ by squarefree integers with sufficiently many prime
factors.
