  In a previous paper, we proved that over a finite field $k$ of sufficiently
large cardinality, all curves of genus at most 3 over k can be modeled by a
bivariate Laurent polynomial that is nondegenerate with respect to its Newton
polytope. In this paper, we prove that there are exactly two curves of genus at
most 3 over a finite field that are not nondegenerate, one over F_2 and one
over F_3. Both of these curves have remarkable extremal properties concerning
the number of rational points over various extension fields.
