  We determine the $166\,104$ extremal monomials of the discriminant of a
quaternary cubic form. These are in bijection with $D$-equivalence classes of
regular triangulations of the $3$-dilated tetrahedron. We describe how to
compute these triangulations and their $D$-equivalence classes in order to
arrive at our main result. The computation poses several challenges, such as
dealing with the sheer amount of triangulations effectively, as well as
devising a suitably fast algorithm for computation of a $D$-equivalence class.
