  Let C be a supersingular genus-2 curve over an algebraically closed field of
characteristic 3. We show that if C is not isomorphic to the curve y^2 = x^5 +
1 then up to isomorphism there are exactly 20 degree-3 maps phi from C to the
elliptic curve E with j-invariant 0. We study the coarse moduli space of
triples (C,E,phi), paying particular attention to questions of rationality. The
results we obtain allow us to determine, for every finite field k of
characteristic 3, the polynomials that occur as Weil polynomials of
supersingular genus-2 curves over k.
