  Let q be a power of a prime and E be an elliptic curve defined over F_q. In
"Combinatorial aspects of elliptic curves" [17], the present author examined a
sequence of polynomials which express the N_k's, the number of points on E over
the field extensions F_{q^k}, in terms of the parameters q and N_1 = #E(F_q).
These polynomials have integral coefficients which alternate in sign, and a
combinatorial interpretation in terms of spanning trees of wheel graphs. In
this sequel, we explore further ramifications of this connection. In
particular, we highlight a relationship between elliptic curves and chip-firing
games on graphs by comparing the groups structures of both. As a coda, we
construct a cyclic rational language whose zeta function is dual to that of an
elliptic curve.
