There is a notion of an optimal quotient
such that
is minimal in its isogeny class. One derives different
optimal quotients of
,
, and from the Jacobians of
Shimura curves. It would be very interesting to have algorithms for
computing structural isomorphism invariants which distinguish these
quotients. See work of Glenn Stevens [Ste89] for a
conjectural answer in the case of elliptic curves (and recent work
of Nike Vatsal (and Stein-Watkins [SW04]) for proofs
of Stevens' conjectures.

Remark 8.7.2 (From Mark Watkins)
This only makes sense (in general) over
, as else you can have
that neither an isogeny or its dual is étale.