For a prime p>2 and q=p^n, we compute a finite generating set for the
SL_2(F_q)-invariants of the second symmetric power representation, showing the
invariants are a hypersurface and the field of fractions is a purely
transcendental extension of the coefficient field. As an intermediate result,
we show the invariants of the Sylow p-subgroups are also hypersurfaces.