Optimal quotients and surjections of Mordell–Weil groups

Answering a question of Ed Schaefer, we show that if J is the Jacobian of a curve C over a number field, if s is an automorphism of J coming from an automorphism of C, and if u   lies in Z[s]⊆EndJ and has connected kernel, then it is not necessarily the case that u gives a surjective map from the Mordell–Weil group of J to the Mordell–Weil group of its image.