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.
Categories: math.AG math.RA