The Group ${\rm Gal}(k_3^{(2)}|k)$ for $k=\mathbb{Q}(\sqrt{-3}, \sqrt{d})$ of Type (3,3)
Let d>0 denote the discriminant of a real quadratic field. For all bicyclic biquadratic fields k=ℚ(−3,d), having a 3-class group of type (3,3), the possibilities for the isomorphism type of the Galois group G=Gal(k3(2)|k) of the second Hilbert 3-class field k3(2) of k are determined. For each coclass graph 𝒢(3,r), r≥1, in the sense of Eick, Leedham-Green, Newman and O’Brien, the roots G of even branches of exactly one coclass tree and, in the case of even coclass cc(G)=r, additionally their siblings of depth 1 and defect 1, turn out to be admissible. The principalization type ϰ(k) of 3-classes of k in its four unramified cyclic cubic extensions K1,…,K4 is given by (0,0,0,0) for cc(G)=1, and by (0,0,4,3) for cc(G)≥2. The theory is underpinned by an extensive numerical verification for all 930 fields k with values of d in the range 0<d<5⋅104, which supports the assumption that all admissible vertices G will actually be realized as Galois groups Gal(k3(2)|k) for certain fields k, asymptotically.