The invariants of the second symmetric power representation of SL2(Fq)

For a prime p>2 and q=pn, we compute a finite generating set for the SL2(Fq)-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.