= Rational torsion of a diagonal cubic
{title2=$E_d(\mathbb Q)_{\mathrm{tors}}$}
For $d\in\mathbb Q^\times$ the rational <torsion subgroup> of $X^3+Y^3+dZ^3=0$ is cyclic of order $3$ if $d$ is a cube, cyclic of order $2$ if $d/2$ is a cube, and trivial otherwise.
To bound torsion, use the <Weierstrass equation of an elliptic curve> $y^2=x^3-432d^2$. At every good <prime> $\ell\equiv2\pmod3$, cubing is a bijection on $\mathbb F_\ell$, so the number of points is $\ell+1$. Reduction injects the prime-to-$\ell$ torsion, since multiplication by an integer prime to $\ell$ is invertible in the <formal group of an elliptic curve>. For any prime $q>3$, the <Dirichlet theorem on primes in arithmetic progressions> supplies a good $\ell\equiv2\pmod3$, $\ell\equiv1\pmod q$, excluding $q$-torsion. Good primes $\ell\equiv5\pmod{12}$ exclude order $4$, and good primes $\ell\equiv11\pmod{18}$ exclude order $9$. Thus every rational torsion point is killed by $6$.
The <flexes of a diagonal cubic> give rational three-torsion exactly when $d$ is a cube, and then it has order $3$. Rational two-torsion requires $x^3=432d^2$, whose sole possible rational root exists exactly when $d/2$ is a cube. This equivalence follows by comparing prime valuations modulo $3$, including the valuation at $2$. The two conditions cannot hold together because $2$ is not a rational cube. The asserted classification follows.
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