Let K = Q(\sqrt{d}) be an imaginary quadratic number field with discriminant d and K_{ur} its maximal unramified extension. We denote by G the Galois group Gal(K_{ur}/K) and l be the length of the class field tower of K: K = K_{0} ⫅ K_{1} ⫅ K_{2} ⫅ ⋯ (K_{i+1} is the Hilbert class field of K_{i}), that is, l is the smallest number with K_{l+1} = K_{l}.

Under GRH we have determined the structure of G for |d| < 1000 except for d = −883, −907, −947, and −996. For |d| ≦ 467, d ≠ −427, the results are unconditional. We give a table of G for K with K_{ur} ≠ K_{1} below. For this table we omit the fields Q(\sqrt{−883}), Q(\sqrt{−907}), and Q(\sqrt{−947}), for which we expect K_{ur} = K_{1} by some reason.

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{(5 + 2\sqrt{−3})(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}

Continued (under GRH)

−d Cl(K)  K_{1}  K_{2}  l  G
472  C_{6} K(\sqrt{2}, α_{4})  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 Q_{12} × C_{2}
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 Q_{20} × C_{2}
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  D_{3} × C_{4}
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  64Γ_{3}p
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_{13}) 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  64Γ_{8}c_{2}
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_{1})  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  64Γ_{3}n_{2}
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  64Γ_{8}e
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_{1})  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 or 4  ?

Now we explain the notations in our table. In the simple expressions of K_{1} and K_{2}, α_{i}, β_{i} and γ_{i} denote any algebraic numbers generating the ith cubic number field of signature (1, 1), the ith quartic number field of signature (2, 1) with Galois group isomorphic to S_{4}, and the ith quintic number field of signature (1, 2) with Galois group isomorphic to D_{5}, respectively, where we consider that the number fields of each signature and each type (of Galois group of normal closure) are numbered up to conjugacy by absolute values of discriminants. (Here we do not need to consider nonisomorphic fields with same discriminants.)

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). For some 2-groups we use designations given in the table due to M. Hall and J. K. Senior.

Fields with |d| < 1000, d ≠ −883, −907, −947 can be classified simply as follows:

{ \begin{matrix}d_{E} ∤ d\{ \\ d_{E} | d\{ \end{matrix}

d ≠ −856, −984, −996 \{ \begin{matrix}h(K_{g}) = \frac{h(K)}{[K_{g}:K]} \{ & \begin{matrix}h(K) = 1 \\ h(K) > 1 \end{matrix} \\ h(K_{g}) > \frac{h(K)}{[K_{g}:K]} \end{matrix} \begin{matrix}⋯ & K_{ur} = K \\ ⋯ & K_{ur} = K_{1} \\ ⋯ & K_{ur} = K_{2} \end{matrix} \}K_{ur} = (K_{g})_{1}

d = −856, −984, −996

d = d_{E} { d = −p prime

d composite

d = d′d_{E} (d′ fundamental quadratic discriminant)

\begin{matrix}⋯ & l ≧ 3 \\ ⋯ & K_{ur} = K_{3} \\ ⋯ & l ≧ 2 \\ ⋯ & l ≧ 3 \end{matrix} \} K_{ur} ⫌ (K_{g})_{1}

Here, we denote by K_{g} the genus field of K. We mean by d_{E} | d that the discriminant d of K is divisible by the discriminant of some quartic number field E.