  Given an elliptic curve C, we study here $N_k = #C(F_{q^k})$, the number of
points of C over the finite field F_{q^k}. This sequence of numbers, as k runs
over positive integers, has numerous remarkable properties of a combinatorial
flavor in addition to the usual number theoretical interpretations. In
particular we prove that $N_k = - W_k(q, - N_1)$ where W_k(q,t) is a
(q,t)-analogue of the number of spanning trees of the wheel graph. Additionally
we develop a determinantal formula for N_k where the eigenvalues can be
explicitly written in terms of q, N_1, and roots of unity. We also discuss here
a new sequence of bivariate polynomials related to the factorization of N_k,
which we refer to as elliptic cyclotomic polynomials because of their various
properties.
