Tamagawa number divisibility of central $L$-values of twists of the Fermat elliptic curve

Given any integer $N>1$ prime to $3$, we denote by $C_N$ the elliptic curve $x^3+y^3=N$. We first study the $3$-adic valuation of the algebraic part of the value of the Hasse-Weil $L$-function $L(C_N,s)$ of $C_N$ over $\mathbb{Q}$ at $s=1$, and we exhibit a relation between the $3$-part of its Tate-Shafarevich group and the number of distinct prime divisors of $N$ which are inert in the imaginary quadratic field $K=\mathbb{Q}(\sqrt{-3})$. In the case where $L(C_N,1)\mathrm{\ne }0$ and $N$ is a product of split primes in $K$, we show that the order of the Tate-Shafarevich group as predicted by the conjecture of Birch and Swinnerton-Dyer is a perfect square.