  In this paper we refine recent work due to A. Shankar, A. N. Shankar, and X.
Wang on counting elliptic curves by conductor to the case of elliptic curves
with a rational 2-torsion point. This family is a small family, as opposed to
the large families considered by the aforementioned authors. We prove the
analogous counting theorem for elliptic curves with so-called square-free index
as well as for curves with suitably bounded Szpiro ratios. We note that our
assumptions on the size of the Szpiro ratios is less stringent than would be
expected by the naive generalization of their approach.
