Flexes of a diagonal cubic ($E_d[3]$)

Let $r^3=d\ne 0$ and let $\zeta$ be a primitive cube root of unity. The nine inflection points of a plane cubic on $X^3+Y^3+d{Z}^3=0$ areThe Hessian criterion for flexes of a plane cubic gives the Hessian determinant $216dXYZ$, so every flex lies on one of the three coordinate lines. Their intersections give precisely the displayed points. With $(1:-1:0)$ as identity, these are exactly the three-torsion points are flexes of a plane cubic configurations: the tangent divisor at a flex is $3P$, giving $3(P-O)=0$.

Over $\mathbb{Q}$ only the first point with $j=0$ is rational on $Z=0$. The other two coordinate lines each give one rational point precisely when $d$ is a rational cube. Consequently the rational three-torsion subgroup is $\mathbb{Z}/3\mathbb{Z}$ when $d$ is a cube and is trivial otherwise.