  Following Katz-Sarnak, Iwaniec-Luo-Sarnak, and Rubinstein, we use the 1- and
2-level densities to study the distribution of low lying zeros for
one-parameter rational families of elliptic curves over Q(t). Modulo standard
conjectures, for small support the densities agree with Katz and Sarnak's
predictions. Further, the densities confirm that the curves' L-functions behave
in a manner consistent with having r zeros at the critical point, as predicted
by the Birch and Swinnerton-Dyer conjecture. By studying the 2-level densities
of some constant sign families, we find the first examples of families of
elliptic curves where we can distinguish SO(even) from SO(odd) symmetry.
