  Let $a$ be an integer which is not of the form $n^2$ or $-3 n^2$ for $n\in
\mathbb{Z}$. Let $E_a$ be the elliptic curve with rational $3$-isogeny defined
by $E_a:y^2=x^3+a$, and $K:=\mathbb{Q}(\mu_3)$. Assume that the $3$-Selmer
group of $E_a$ over $K$ vanishes. It is shown that there is an explicit
infinite set of cubefree integers $m$ such that the $3$-Selmer groups over $K$
of $E_{m^2 a}$ and $E_{m^4 a}$ both vanish. In particular, the ranks of these
cubic twists are seen to be $0$ over $K$. Our results are proven by studying
stability properties of $3$-Selmer groups in cyclic cubic extensions of $K$,
via local and global Galois cohomological techniques.
