Quotients of $J_0(p)^-$

We computed the quantities $m_A$, $m_X$, and $\Phi_X$ for each abelian variety $A_f$ associated to a newform of prime level $p$ with $p\leq 631$. Table 2 lists those $A_f$ for which $w_p=-1$, along with the order of the corresponding component group. The first column, which is labeled ``$A$'' contains a description of $A_f$, the second column, labeled ``$d$'', contains the dimension of $A_f$, and the third column, labeled ``$\char93 \Phi_A$'', contains the order $\char93 \Phi_{A_f,p}(\overline{\mathbf{F}}_p)$ of the component group.

Remark 7.4   Theorem 7.2 together with [13, Prop. II.17.10] imply that the component groups of the $A_f$ for which $w_p=+1$ are trivial, so we do not list them. An optimal quotient $A_f$ of $J_0(p)$ with nonzero component group has nonzero rational torsion (by Theorem 7.2), so it factors through the Eisenstein quotient of $J_0(p)$. Also $w_p$ acts as $-1$ on the Eisenstein quotient of $J_0(p)$, which is [13, Prop. II.17.10], and which is a deep result because of subtleties at the prime 2 (see the discussion in [13, III.1]).

Table 1: Component groups of quotients of $J_0(N)$

Table 2: Component groups of quotients of $J_0(p)^{-}$