  We give a moduli interpretation to the quotient of (nondegenerate) binary
cubic forms with respect to the natural $\text{GL}_2$-action on the variables.
In particular, we show that these $\text{GL}_2$ orbits are in bijection with
pairs of $j$-invariant $0$ elliptic curves together with $3$-torsion Brauer
classes that are invariant under complex multiplication. The binary cubic
generic Clifford algebra plays a key role in the construction of this
correspondence.
