Table of imaginary quadratic number fields K = Q(\sqrt{d}), |d| < 1000 with K_{ur} ≠ K_{1}

−d Cl(K)  K_{1}  K_{2}  l  G
115  C_{2}  K(\sqrt{5})  K_{1}(α_{1})  2  D_{3}
120  V_{4}  K(\sqrt{−3}, \sqrt{5})  K_{1}(\sqrt{(2\sqrt{2} + \sqrt{5})(2 + \sqrt{5})})  2  Q_{8}
155  C_{4}  K(\sqrt{(−1 + 5\sqrt{5})/2})  K_{1}(α_{2})  2 Q_{12}
184  C_{4}  K(\sqrt{−3 + 4\sqrt{2}})  K_{1}(α_{1})  2 Q_{12}
195  V_{4}  K(\sqrt{−3}, \sqrt{5})  2 Q_{16}
235  C_{2}  K(\sqrt{5})  K_{1}(γ_{1})  2  D_{5}
248  C_{8}  K_{1}(α_{2})  2  I_{3}^{8}
255 C_{6} × C_{2} K(\sqrt{5}, \sqrt[3]{(9 + \sqrt{85})/2}) K_{1}(\sqrt{(2\sqrt{−3} + \sqrt{5})(2 + \sqrt{5})}) 2 Q_{8} × C_{3}
260 C_{4} × C_{2} K(\sqrt{5}, \sqrt{8 + \sqrt{65}})  2 M_{16}
276 C_{4} × C_{2} K(\sqrt{−1}, \sqrt{13 + 8\sqrt{3}}) K_{1}(α_{1})  2 Q_{12} × C_{2}
280  V_{4}  K(\sqrt{−7}, \sqrt{5})  2 Q_{16}
283  C_{3}  K(α_{31})  K_{1}(β_{1})  3  Ã_{4}
295  C_{8}  K_{1}(α_{4})  2  I_{3}^{8}
299  C_{8}  K_{1}(α_{1})  2  I_{3}^{8}
312  V_{4}  K(\sqrt{−3}, \sqrt{2})  2 Q_{16}
331  C_{3}  K(α_{36})  K_{1}(β_{2})  3  Ã_{4}
340  V_{4}  K(\sqrt{−1}, \sqrt{5})  2 SD_{16}
355  C_{4}  K(\sqrt{−3 + 4\sqrt{5}})  2 Q_{28}
372  V_{4}  K(\sqrt{−1}, \sqrt{−3})  K_{1}(α_{2})  2  D_{6}
376  C_{8}  K_{1}(γ_{1})  2  I_{5}^{8}
391  C_{14}  K_{1}(α_{1})  2 D_{3} × C_{7}
395  C_{8}  K_{1}(γ_{2})  2  I_{5}^{8}
403  C_{2}  K(\sqrt{13})  K_{1}(α_{2})  2  D_{3}
408  V_{4}  K(\sqrt{−3}, \sqrt{2})  K_{1}(\sqrt{−(5 + \sqrt{17})/2})  2  D_{4}
415  C_{10} K(\sqrt{5}, γ_{18})  K_{1}(α_{6})  2 D_{3} × C_{5}
420  C_{2}^{3}  K(\sqrt{−1}, \sqrt{−3}, \sqrt{5})  2 32Γ_{4}c_{3}
435  V_{4}  K(\sqrt{−3}, \sqrt{5})  2 Q_{16} ⋉ C_{3}
440 C_{6} × C_{2} K(\sqrt{2}, \sqrt{5}, α_{50})  2 Q_{16} × C_{3}
455 C_{10} × C_{2} K(\sqrt{−7}, \sqrt{5}, γ_{21})  K_{1}(\sqrt{(2\sqrt{13} − 3\sqrt{5})(2 + \sqrt{5})}) 2 Q_{8} × C_{5}

Except for K = Q(\sqrt{−427}), Gal(K_{ur}/K) is determined unconditionally for |d| ≦ 463. For K = Q(\sqrt{−427}), K_{1} is the top of the class field tower, however, the possibility that Gal(K_{ur}/K_{1}) ≅ PSL(2, 7) is eliminated under GRH.

Continued(under GRH)

−d Cl(K)  K_{1}  K_{2}  l  G
472  C_{6} K(\sqrt{2}, α_{54})  K_{1}(α_{4}) 2  D_{3} × C_{3}
483  V_{4}  K(\sqrt{−3}, \sqrt{−7})  K_{1}(α_{1}) 2  D_{6}
491  C_{9}  K_{1}(β_{3}) 3  Ã_{4} ⋏ C_{9}
515  C_{6} K(\sqrt{5}, α_{60})  K_{1}(γ_{3}) 2  D_{5} × C_{3}
520  V_{4}  K(\sqrt{−2}, \sqrt{5})  2  Q_{24}
527  C_{18}  K_{1}(α_{2}) 2  D_{3} × C_{9}
535  C_{14}  K_{1}(α_{9}) 2  D_{3} × C_{7}
552 C_{4} × C_{2} K(\sqrt{−3}, \sqrt{−1 + 2\sqrt{6}})  K_{1}(α_{1}) 2  D_{3} × C_{4}
555  V_{4}  K(\sqrt{−3}, \sqrt{5})  2  Q_{32}
563  C_{9}  K_{1}(β_{4}) 3  Ã_{4} ⋏ C_{9}
564 C_{4} × C_{2} K(\sqrt{−1}, \sqrt{1 + 4\sqrt{3}})  K_{1}(γ_{1}) 2  D_{5} × C_{4}
568  C_{4} K(\sqrt{−1 + 6\sqrt{2}})  2  Q_{28}
580 C_{4} × C_{2} K(\sqrt{5}, \sqrt{12 + \sqrt{145}})  2 32Γ_{3}f ⋉ C_{3}
595  V_{4}  K(\sqrt{−7}, \sqrt{5})  2  Q_{40}
611  C_{10} K(\sqrt{13}, γ_{28})  K_{1}(γ_{1}) 2  D_{5} × C_{5}
632  C_{8}  K_{1}(γ_{2}) 2  I_{5}^{8}
635  C_{10} K(\sqrt{5}, γ_{31})  K_{1}(γ_{5}) 2  D_{5} × C_{5}
643  C_{3} K(α_{72})  K_{1}(β_{5}) 3  Ã_{4}
644 C_{8} × C_{2}  K_{1}(α_{1}) 2  D_{3} × C_{8}
651 C_{4} × C_{2} K(\sqrt{−7}, \sqrt{(13 + \sqrt{217})/2})  K_{1}(α_{2}) 2 Q_{12} × C_{2}
655  C_{12} K(\sqrt{7 + 6\sqrt{5}}, α_{75})  K_{1}(γ_{6}) 2 Q_{20} × C_{3}
660  C_{2}^{3} K(\sqrt{−1}, \sqrt{−3}, \sqrt{5})  2  64Γ_{15}f_{2}
663 C_{8} × C_{2}  2  Γ_{3,2}^{2}
664  C_{10} K(\sqrt{2}, γ_{32})  K_{1}(α_{6}) 2  D_{3} × C_{5}
667  C_{4} K(\sqrt{(−13 + 3\sqrt{29})/2})  K_{1}(α_{1}) 2  Q_{12}
680 C_{6} × C_{2} K(\sqrt{−2}, \sqrt{5}, α_{80})  2 Q_{16} × C_{3}
687  C_{12} K(\sqrt{(11 + \sqrt{229})/2}, \sqrt[3]{(27 + \sqrt{229})/2})  3 (S_{4} ⋏ C_{4}) × C_{3}
695  C_{24}  K_{1}(α_{4}) 2  I_{3}^{8} × C_{3}
696 C_{6} × C_{2} K(\sqrt{2}, \sqrt[3]{99 + 13\sqrt{58}})  2 Q_{24} × C_{3}
708  V_{4}  K(\sqrt{−1}, \sqrt{−3})  K_{1}(α_{4}) 2  D_{6}
715  V_{4}  K(\sqrt{−11}, \sqrt{5})  2 Q_{16} ⋉ C_{5}
728 C_{6} × C_{2} K(\sqrt{−7}, \sqrt{2}, α_{86})  2 Q_{16} × C_{3}
731  C_{12} K(\sqrt{−5 + 2\sqrt{17}}, α_{87})  3  Ã_{4} ⋎ C_{8}
740 C_{8} × C_{2}  2  M_{32}
744 C_{6} × C_{2} K(\sqrt{−2}, \sqrt[3]{63 + 8\sqrt{62}})  K_{1}(α_{2}) 2  D_{3} × C_{6}
751  C_{15} K(α_{91}, γ_{39})  K_{1}(β_{8}) 3  Ã_{4} × C_{5}
755  C_{12} K(\sqrt{13 + 8\sqrt{5}}, α_{92})  2 Q_{28} × C_{3}

Continued(under GRH)

−d Cl(K)  K_{1}  K_{2}  l  G
759 C_{12} × C_{2} K(\sqrt{−3}, \sqrt{(−5 + \sqrt{69})/2}, α_{94}) K_{1}(α_{1})  2  D_{3} × C_{12}
760  V_{4}  K(\sqrt{2}, \sqrt{5})  2  Q_{32} ⋉ C_{3}
767  C_{22}  K_{1}(α_{4})  2  D_{3} × C_{11}
771  C_{6}  K(\sqrt[3]{16 + \sqrt{257}})  3  S_{4} × C_{3}
795  V_{4}  K(\sqrt{−3}, \sqrt{5})  2  Q_{40}
799  C_{16}  K_{1}(γ_{1})  2  I_{5}^{16}
820 C_{4} × C_{2} K(\sqrt{−1}, \sqrt{6 + \sqrt{41}})  2  Γ_{2,3}^{1}
824  C_{20} K(\sqrt{5 + 8\sqrt{2}}, γ_{46})  K_{1}(γ_{3})  2  Q_{20} × C_{5}
835  C_{6}  K(\sqrt{5}, α_{102})  2  D_{11} × C_{3}
840  C_{2}^{3}  K(\sqrt{−2}, \sqrt{−3}, \sqrt{5})  2  32Γ_{4}d
851  C_{10} K(\sqrt{−23}, γ_{49})  K_{1}(α_{4})  2  D_{3} × C_{5}
852 C_{4} × C_{2} K(\sqrt{−1}, \sqrt{2 + 5\sqrt{3}})  2  Q_{28} × C_{2}
856  C_{6}  K(\sqrt{2}, α_{105})  K_{1}(α_{9})  4  S̃_{4} × C_{3}
868 C_{4} × C_{2} K(\sqrt{−1}, \sqrt{9 + 4\sqrt{7}})  K_{1}(α_{2})  2  Q_{12} × C_{2}
884 C_{8} × C_{2}  2  Γ_{3,2}^{1}
888 C_{6} × C_{2} K(\sqrt{2}, \sqrt[3]{43 + 5\sqrt{74}})  2  Q_{32} × C_{3}
895  C_{16}  K_{1}(γ_{9})  2  I_{5}^{16}
899  C_{14}  K_{1}(α_{2})  2  D_{3} × C_{7}
915 C_{4} × C_{2} K(\sqrt{−3}, \sqrt{(−1 + \sqrt{61})/2})  2  Γ_{2,3}^{2}
916  C_{10} K(\sqrt{−1}, γ_{52})  3  S_{4} × C_{5}
920 C_{10} × C_{2} K(\sqrt{2}, \sqrt{5}, γ_{53})  2 (Q_{16} ⋉ C_{3}) × C_{5}
923  C_{10} K(\sqrt{13}, γ_{54})  2  D_{7} × C_{5}
935 C_{14} × C_{2}  2  Q_{16} × C_{7}
943  C_{16}  K_{1}(α_{4})  2  I_{3}^{16}
948 C_{6} × C_{2} K(\sqrt{−1}, \sqrt[3]{80 + 9\sqrt{79}})  K_{1}(γ_{2})  2  D_{5} × C_{6}
952 C_{4} × C_{2} K(\sqrt{−7}, \sqrt{−9 + 10\sqrt{2}})  2 (C_{4} ≀ C_{2}) ⋉ C_{5}
955 C_{4} K(\sqrt{(−9 + 13\sqrt{5})/2}) 2 Q_{52}
984 C_{6} × C_{2} K(\sqrt{2}, \sqrt[3]{9 + \sqrt{82}})  3 (Q_{16} ⋉ C_{3}^{2}) × C_{3}
987 C_{4} × C_{2} K(\sqrt{−3}, \sqrt{(−1 + 3\sqrt{21})/2})  K_{1}(γ_{1})  2  Q_{20} × C_{2}
995  C_{8}  2  I_{9}^{8}
996 C_{6} × C_{2} K(\sqrt{−1}, \sqrt[3]{82 + 9\sqrt{83}})  K_{1}(α_{2}) ≧ 3  ?

Supplements.

d = −520 : K_{2} = K_{1}(\sqrt{(2\sqrt{−2} + \sqrt{5})(2 + \sqrt{5})}, α_{8})

For d = −883, −907, −947, K_{1} is the top of the class field tower, however, the possibility that Gal(K_{ur}/K_{1}) ≅ PSL(2, 7) is not still eliminated. We expect K_{ur} = K_{1}.

For d = −996, K has an unramified S_{4} × C_{6}-extension. We expect that this extension coincides with K_{ur}.

As usual, C_{n} is the cyclic group of order n, V_{4} is the four group, that is, V_{4} = C_{2}^{2} = C_{2} × C_{2}, D_{n}(n ≧ 3) is the dihedral group of order 2n, Q_{4n}(n ≧ 2) is the generalized quaternion group of order 4n, and SD_{8n}(n ≧ 2) is the semi-dihedral group of order 8n:

\begin{aligned}D_{n} & = ⟨ a, b | a^{n} = b^{2} = 1, b^{−1}ab = a^{−1} ⟩, \\ Q_{4n} & = ⟨ a, b | a^{2n} = 1, b^{2} = a^{n}, b^{−1}ab = a^{−1} ⟩, \\ SD_{8n} & = ⟨ a, b | a^{4n} = b^{2} = 1, b^{−1}ab = a^{2n−1} ⟩. \end{aligned}

I_{n}^{2m}(m ≧ 2, n ≧ 3) denotes the group of order 2mn given by

⟨ a, b | a^{2m} = b^{n} = 1, a^{−1}ba = b^{−1} ⟩.

M_{2^{n}}(n ≧ 4) denotes the modular group of order 2^{n} given by

⟨ a, b | a^{2^{n−1}} = b^{2} = 1, b^{−1}ab = a^{2^{n−2}+1} ⟩.

Ã_{4} is the double cover of A_{4} : Ã_{4} ≅ SL(2, 3). S̃_{4} is the double cover of S_{4} with S̃_{4} ≅ GL(2, 3). For some 2-groups we use designations given in the table by M. Hall and J. K. Senior. For simplicity, for some of them we use the following designations. Γ_{m,t}^{1} denotes the group of order 2^{m+t+1} given by

⟨ a, b | a^{4} = b^{2^{m}} = 1, c = [a, b], a^{2} = c^{2^{t−1}}, [a, c] = c^{2}, [ab, c] = 1 ⟩.

Γ_{m,t}^{2} denotes the group of order 2^{m+t+1} given by

⟨ a, b | a^{4} = b^{2^{m+1}} = 1, c = [a, b], a^{2} = b^{2^{m}} = c^{2^{t−1}}, [a, c] = c^{2}, [ab, c] = 1 ⟩.

We note that Γ_{3,2}^{2} = 64Γ_{3}p, Γ_{2,3}^{1} = 64Γ_{8}c_{2}, Γ_{3,2}^{1} = 64Γ_{3}n_{2}, Γ_{2,3}^{2} = 64Γ_{8}e, Γ_{2,2}^{2} = 32Γ_{3}e (≅ C_{4} ≀ C_{2}).