The Capitulation Problem in Certain Pure Cubic Fields

Let $\Gamma=\mathbb{Q}(\sqrt[3]{n})$ be a pure cubic field with normal closure $k=\mathbb{Q}(\sqrt[3]{n},\zeta)$, where $n>1$ denotes a cube free integer, and $\zeta$ is a primitive cube root of unity. Suppose $k$ possesses an elementary bicyclic $3$-class group $\mathrm{Cl}_3(k)$, and the conductor of $k/\mathbb{Q}(\zeta)$ has the shape $f\in\lbrace pq_1q_2,3pq,9pq\rbrace$ where $p\equiv 1\,(\mathrm{mod}\,9)$ and $q,q_1,q_2\equiv 2,5\,(\mathrm{mod}\,9)$ are primes. It is disproved that there are only two possible capitulation types $\varkappa(k)$, either type $\mathrm{a}.1$, $(0000)$, or type $\mathrm{a}.2$, $(1000)$. Evidence is provided, theoretically and experimentally, of two further types, $\mathrm{b}.10$, $(0320)$, and $\mathrm{d}.23$, $(1320)$.