  Let $N$ and $p$ be primes $\geq 5$ such that $p \mid \mid N-1$. In this
situation, Mazur defined and studied the $p$-Eisenstein quotient
$\tilde{J}^{(p)}$ of $J_0(N)$. We prove a kind of modulo $p$ version of the
Birch and Swinnerton-Dyer conjecture for the ``$p$-Eisenstein part'' of even
quadratic twists of $\tilde{J}^{(p)}$. Our result is the analogue for even
quadratic twists of a result of Mazur concerning odd quadratic twists.
