Optimal quotients of Jacobians with toric reduction and component groups

Let J be a Jacobian variety with toric reduction over a local field K. Let J → E be an optimal quotient defined over K, where E is an elliptic curve. We give examples in which the functorially induced map φJ → φ_E on component groups of the Neron models is not surjective. This answers a question of Ribet and Takahashi. We also give various criteria under which φJ → φ_E E is surjective and discuss when these criteria hold for the Jacobians of modular curves.