Hilbert's 10th Problem via Mordell curves

We show that for $5\mathrm{/}6$-th of all primes $p$, Hilbert’s 10-th Problem is unsolvable for $\mathbb{Q}({\zeta }_3,\sqrt[3]{p})$. We also show that there is an infinite set $S$ of square free integers such tha Hilbert’s 10-th Problem is unsolvable over the number fields $\mathbb{Q}({\zeta }_3,\sqrt{D},\sqrt[3]{p})$ for every $D\in S$ and every prime $p\equiv 2,5(\mathrm{m}\mathrm{o}\mathrm{d}9)$. We use the CM elliptic curves $Y^2=X^3-432D^2$ associated to the cube sum problem, with $D$ varying in suitable congruence class, in our proof.