  For a quadratic endomorphism of the affine line defined over the rationals,
we consider the problem of bounding the number of rational points that
eventually land at the origin after iteration. In the article ``Uniform Bounds
on Pre-Images Under Quadratic Dynamical Systems,'' by two of the present
authors and five others, it was shown that the number of rational iterated
pre-images of the origin is bounded as one varies the morphism in a certain
one-dimensional family. Subject to the validity of the Birch and
Swinnerton-Dyer conjecture and some other related conjectures for the L-series
of a specific abelian variety and using a number of modern tools for locating
rational points on high genus curves, we show that the maximum number of
rational iterated pre-images is six. We also provide further insight into the
geometry of the ``pre-image curves.''
