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N.Garc\'ia-Fritz and H.Pasten showed that Hilbert's 10th problem isunsolvable in the ring of integers of number fields of the form$\mathbb{Q}(\sqrt[3]{p},\sqrt{-q})$ for positive proportions of primes $p$ and$q$. We improve their proportions and extend their results to the case ofnumber fields of the form $\mathbb{Q}(\sqrt[3]{p},\sqrt{Dq})$, where $D$belongs to an explicit family of positive square-free integers. We achieve thisby using multiple elliptic curves, and replace their Iwasawa theory argumentsby a more direct method.

Studying Hilbert's 10th problem via explicit elliptic curves