  In this paper we suggest a definition for the category of mixed motives
generated by the motive h^1(E) for E an elliptic curve without complex
multiplication. We then compute the cohomology of this category. Modulo a
strengthening of the Beilinson-Soule conjecture, we show that the cohomology of
our category agrees with the expected motivic cohomology groups. Finally for
each pure motive (Sym^{n}h^1(E))(-1) we construct families of nontrivial
motives whose highest associated weight graded piece is $Sym^{n}h^1(E))(-1).
  This paper was essentially written in the late 1990's whilst the author was
at the University of Chicago. The author apologizes for the tardiness of this
posting, and hopes the reader will still find the content interesting.
