(A preprint is available online.)

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*.