= Flexes of a diagonal cubic
{title2=$E_d[3]$}
Let $r^3=d\ne0$ and let $\zeta$ be a primitive cube root of unity. The nine <inflection points of a plane cubic> on $X^3+Y^3+dZ^3=0$ are
$$
(1:-\zeta^j:0),\qquad(0:-r\zeta^j:1),\qquad(-r\zeta^j:0:1),\qquad j=0,1,2.
$$
The <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.
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