The Capitulation Problem in Certain Pure Cubic Fields

Let $\mathrm{\Gamma }=\mathbb{Q}(\sqrt[3]{n})$ 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 ${\mathrm{C}\mathrm{l}}_3(k)$, and the conductor of $k\mathrm{/}\mathbb{Q}(\zeta )$ has the shape $f\in \{p{q}_1{q}_2,3pq,9pq\}$ where $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}9)$ and $q,q_1,q_2\equiv 2,5(\mathrm{m}\mathrm{o}\mathrm{d}9)$ are primes. It is disproved that there are only two possible capitulation types $\mathrm{\varkappa }(k)$, 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 $\mathrm{d}.23$, $(1320)$.