On the arithmetic of special values of $L$-functions for certain abelian varieties with a rational isogeny

Let $N$ and $p$ be primes $\ge 5$ 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.