$C/ \Bbb{Q}: 3X^3 + 4Y^3 + 5Z^3 = 0$ is known to be a nontrivial element of the Tate–Shafarevich group of the elliptic curve $E/\Bbb{Q}:X^3 + Y^3 + 60Z^3 = 0$. It is also an example of an abelian variety for which finiteness of Sha is known. In fact, $|\mathrm{III}(E/\Bbb{Q})| = 3^2$.


But I have never seen the proof of $|\mathrm{III}(E/\Bbb{Q})| = 3^2$.I don't either know the proof of $[C]$ has order $3$ in $\mathrm{III}(E/\Bbb{Q})$.
Maybe this is first proved by Rubin, but I cannot find article which discusses this example.
Could you give me an reference for the proof?



Tate–Shafarevich group of Jacobian of Selmer curve $3X^3 + 4Y^3 + 5Z^3 = 0$
$\DeclareMathOperator{\sha}{Ш}$
I am not sure that the proof that Sha has order 9 is anywhere spelled out in full. Here the ideas how to do it.


First, that the order of $C$ is three in the $\sha$ is just saying that it has a point over a field of degree 3 (index=period), which is obvious, and none of degree 1, which was first proven by Selmer. See Cassel's lectures, chapter 18.


A $2$-descent, proves that $E(\mathbb{Q})$ has rank $0$ and that $\sha[2]$ is trivial.
Calculating the torsion subgroup of $E$ (trivial), the Tamagawa numbers (all trivial and the value of $L(E,1)$, either using cm-theory or modular symbols, reveals that the Birch and Swinnerton-Dyer conjecture is equivalent to $\sha$ having $9$ elements.


Rubin's work on the main conjecture of elliptic curves with CM, as in "Tate-Shafarevich groups and $L$-functions of elliptic curves with complex multiplication" for instance, shows that the order of the $p$-primary part of $\sha$ is correctly predicted by BSD for all primes not dividing the order of the units in $\mathbb{Q}(\sqrt{-3})$. For our curve this means $\sha[p]=0$ for all primes $p>3$. But this does not cover the very bad prime $3$; though maybe has done this since, I don't know.


Selmer's original work proved that $\sha[3]$ contains $9$ elements, by doing a second descent $E\to E'$ via an isogeny of degree $3$. This exact example appears in magma's documentation of its ThreeDescent function: <https://magma.maths.usyd.edu.au/magma/handbook/text/1515#17622> .
More interestingly this calculation also shows that the $3$-Selmer group of the curves $E'$ at the other end of the $3$-isogeny is trivial. Hence the $3$-part of BSD is true for that curve and since this is invariant under isogeny, it is also true for ours. Therefore $\sha$ has order $9$ with the structure being $\mathbb{Z}/3\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}$.


This last step should also be possible using Heegner points with $D=-17$. Sage uses this in "prove\_BSD" for the curve $E'$.


