\input amstex
\documentstyle{amsppt}
\define\Q{{\bold Q}}
\define\Gal{{\operatorname{Gal}}}
\define\Cl{{\operatorname{Cl}}}
\define\lt{\ltimes}
\define\rt{\rtimes}
\NoPageNumbers

\mag=\magstep1

\document
Let $K=\Q(\sqrt{d})$ be a real 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 $K$:
$K=K_0\subseteqq K_1\subseteqq K_2\subseteqq\cdots$
($K_{i+1}$ 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 $d\leqq 5565$.
For $d\leqq 2636$, the results are unconditional.
We give a table of $G$ for $K$ with $K_{ur}\ne K_1$ below.

We the following notations in the table.
In the simple expressions of $K_1$ and $K_2$,
$a_i, b_i$ and $c_i$ denote any algebraic numbers generating
the $i$th totally real cubic number field,
the $i$th totally real quartic number field
with Galois group isomorphic to $S_4$, and
the $i$th totally real quintic number field
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.)

\NoBlackBoxes
\vskip.5cm
\def\hlf{\noalign{\hrule}}
\def\vspa{height2pt &\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr}
\def\vspb{height1pt &\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr}
\def\table#1{
\vbox\bgroup\offinterlineskip
\halign\bgroup
% preamble
\vrule## &
\ \hfil $##$ \ \strut & \vrule## & \ \hfil $##$ \hfil
& \vrule## & \ $##$ \hfil \ & \vrule## & \ $##$ \hfil \ 
& \vrule## & \ \hfil $##$ & \vrule## &\ \hfil $##$ \hfil & \vrule##
\cr
%data
\multispan{13}{ \hfil #1 \hfil } \cr
\noalign{\smallskip}
\hlf
%\vspb
& d && \Cl(K) &&
\multispan{1}{\hfil $K_1$ \hfil} &&
\multispan{1}{\hfil $K_2$ \hfil} && l && G & \cr
%\vspb
\hlf
\vspb
}
\def\endtable{\egroup\egroup}
\NoBlackBoxes
\centerline{Table of real quadratic number fields $K=\Q(\sqrt{d})$,
$d\leqq 5565$ with $K_{ur}\ne K_1$}
\vskip-.5cm
$$
\table{}
& 520 && V_4 &&
K(\sqrt{2},\sqrt{5}) &&
K(\sqrt{\smash{(1+\sqrt{2})(3\sqrt{2}+\sqrt{5})}\vphantom{-3+4\sqrt{2}}})
&& 2 && Q_8 & \cr						% 159
& 1145 && C_4 &&
K(\sqrt{\smash{(31+3\sqrt{5})/2}\vphantom{-3+4\sqrt{2}}})
&& K_1(a_2) && 2 && Q_{12} & \cr				% 348
& 1160 && V_4 && K(\sqrt{5},\sqrt{29}) && K_1(\sqrt{7+2\sqrt{5}})
&& 2 && D_4 & \cr						% 353
& 1285 && C_2 && K(\sqrt{5}) && K_1(a_3) && 2 && D_3 & \cr	% 392
& 1580 && C_2 && K(\sqrt{5}) && K_1(a_4) && 2 && D_3 & \cr	% 479
& 1605 && C_2 && K(\sqrt{5}) && K_1(a_5) && 2 && D_3 & \cr	% 487
& 1640 && V_4 && K(\sqrt{2},\sqrt{5}) &&
K_1(\sqrt{7+2\sqrt{2}}) && 2 && D_4 & \cr			% 497
& 1740 && V_4 && K(\sqrt{3},\sqrt{5}) && && 2 && Q_{16} & \cr	% 528
& 1785 && C_4\times C_2 &&
K(\sqrt{5},\sqrt{\smash{(19+\sqrt{21})/2}\vphantom{-3+4\sqrt{2}}})
&& && 2 && M_{16} & \cr						% 544
& 1820 && V_4 && K(\sqrt{5},\sqrt{13}) &&
K(\sqrt{\smash{(\sqrt{13}+2\sqrt{5})(2+\sqrt{5})}\vphantom{-3+4\sqrt{2}}})
&& 2 && Q_8 & \cr						% 553
& 1832 && C_2 && K(\sqrt{2}) && K_1(a_2) && 2 && D_3 & \cr	% 556
& 1885 && V_4 && K(\sqrt{5},\sqrt{13}) && K_1(\sqrt{7+2\sqrt{5}})
&& 2 && D_4 & \cr						% 573
& 1957 && C_3 && K(a_{54}) && K_1(b_1) && 2 && A_4 & \cr	% 594
& 2005 && C_4 &&
K(\sqrt{\smash{(43+7\sqrt{5})/2}\vphantom{-3+4\sqrt{2}}})
&& K_1(c_1) && 2 && Q_{20} & \cr				% 609
& 2056 && C_4 &&
K(\sqrt{17+4\sqrt{2}}) && K_1(a_3) && 2 && Q_{12} & \cr		% 625
& 2120 && V_4 && K(\sqrt{2},\sqrt{5}) &&
K(\sqrt{\smash{(3\sqrt{5}+7\sqrt{2})(1+\sqrt{2})}\vphantom{-3+4\sqrt{2}}})
&& 2 && Q_8 & \cr						% 645
& 2345 && C_2 && K(\sqrt{5}) && K_1(a_7) && 2 && D_3 & \cr	% 712
& 2405 && V_4 && K(\sqrt{5},\sqrt{13}) &&
K(\sqrt{\smash{(4\sqrt{13}+7\sqrt{5})(2+\sqrt{5})}\vphantom{-3+4\sqrt{2}}})
&& 2 && Q_8 & \cr						% 730
& 2460 && V_4 && K(\sqrt{3},\sqrt{5}) &&
K_1(\sqrt{\smash{(13+\sqrt{5})/2}\vphantom{-3+4\sqrt{2}}})
&& 2 && D_4 & \cr						% 747
& 2465 && V_4 && K(\sqrt{5},\sqrt{17}) && && 2 && Q_{16} & \cr	% 749
& 2568 && C_2 && K(\sqrt{2}) && K_1(a_5) && 2 && D_3 & \cr	% 780
\hlf
\endtable
\hfil
$$

\newpage

\vcorrection{-.35in}
$$
\table{Continued (under GRH)}
& 2652 && V_4 && K(\sqrt{3}, \sqrt{13}) &&
K_1(\sqrt{\smash{(9+\sqrt{13})/2}\vphantom{-3+4\sqrt{2}}})
&& 2 && D_4 & \cr						% 806
& 2748 && C_2 && K(\sqrt{3}) && K_1(a_2) && 2 && D_3 & \cr	% 835
& 2777 && C_3 &&
K(a_{80}) && K_1(b_2) && 3 && \widetilde{A_4} & \cr		% 846
& 2840 && C_2 && K(\sqrt{5}) && K_1(a_{10}) && 2 && D_3 & \cr	% 863
& 2856 && V_4 && K(\sqrt{2}, \sqrt{17}) &&
K_1(\sqrt{\smash{3(5+2\sqrt{2})}\vphantom{-3+4\sqrt{2}}})
&& 2 && D_4 & \cr						% 867
& 2885 && C_2 && K(\sqrt{5}) &&  && 2 && D_7 & \cr		% 877
& 2977 && C_2 && K(\sqrt{13}) && K_1(a_2) && 2 && D_3 & \cr	% 905
& 3045 && V_4 && K(\sqrt{5}, \sqrt{21}) && K_1(\sqrt{7+2\sqrt{5}})
&& 2 && D_4 & \cr						% 924
& 3084 && C_2 && K(\sqrt{3}) && K_1(a_3) && 2 && D_3 & \cr	% 938
& 3145 && V_4 && K(\sqrt{5}, \sqrt{17}) &&
K(\sqrt{\smash{(4\sqrt{17}+11\sqrt{5})(2+\sqrt{5})}\vphantom{-3+4\sqrt{2}}})
&& 2 && Q_8 & \cr						% 955
& 3180 && V_4 && K(\sqrt{3}, \sqrt{5}) &&
K(\sqrt{\smash{(\sqrt{53}+4\sqrt{5})(2+\sqrt{5})}\vphantom{-3+4\sqrt{2}}})
&& 2 && Q_8 & \cr						% 967
& 3208 && C_2 && K(\sqrt{2}) && K_1(c_1) && 2 && D_5 & \cr	% 976
& 3341 && C_4 &&
K(\sqrt{33+8\sqrt{13}}) && K_1(a_3) && 2 && Q_{12} & \cr	% 1019
& 3445 && V_4 && K(\sqrt{5}, \sqrt{13}) &&
K_1(\sqrt{23+2\sqrt{13}}) && 2 && D_4 & \cr			% 1048
& 3480 && C_4\times C_2 &&
K(\sqrt{5}, \sqrt{19+6\sqrt{6}}) && K_1(\sqrt{7+2\sqrt{5}})
&& 2 && C_4\ltimes C_4 & \cr					% 1059
& 3485 && V_4 &&
K(\sqrt{5}, \sqrt{17}) && K_1(a_{12}) && 2 && D_6 & \cr		% 1061
& 3560 && V_4 && K(\sqrt{2}, \sqrt{5}) &&
K_1(\sqrt{13+4\sqrt{5}}) && 2 && D_4 & \cr			% 1081
& 3660 && V_4 && K(\sqrt{3}, \sqrt{5}) &&
K_1(\sqrt{\smash{3(9+2\sqrt{5})}\vphantom{-3+4\sqrt{2}}})
&& 2 && D_4 & \cr						% 1113
& 3665 && C_2 && K(\sqrt{5}) && K_1(a_{23}) && 2 && D_3 & \cr	% 1115
& 3720 && V_4 && K(\sqrt{2}, \sqrt{5}) &&
K(\sqrt{\smash{(7\sqrt{2}+\sqrt{5})(1+\sqrt{2})}\vphantom{-3+4\sqrt{2}}})
&& 2 && Q_8 & \cr						% 1131
& 3752 && C_2 && K(\sqrt{2}) && K_1(a_7) && 2 && D_3 & \cr	% 1140
& 3784 && C_2 && K(\sqrt{2}) && K_1(a_8) && 2 && D_3 & \cr	% 1149
& 3805 && C_4 &&
K(\sqrt{29+4\sqrt{5}}) && K_1(a_7) && 2 && Q_{12} & \cr		% 1156
& 3885 && V_4 && K(\sqrt{5}, \sqrt{21}) &&
K_1(\sqrt{\smash{(11+\sqrt{37})/2}\vphantom{-3+4\sqrt{2}}})
&& 2 && D_4 & \cr						% 1182
& 3893 && C_2 && K(\sqrt{17}) && K_1(a_2) && 2 && D_3 & \cr	% 1184
& 3965 && V_4 && K(\sqrt{5}, \sqrt{13}) &&
&& 2 && SD_{16} & \cr						% 1207
& 3981 && C_3 && K(a_{127}) && K_1(b_3) && 2 && A_4 & \cr	% 1212
& 4040 && V_4 && K(\sqrt{2}, \sqrt{5}) &&
K_1(\sqrt{11+2\sqrt{5}}) && 2 && D_4 & \cr			% 1229
& 4060 && V_4 && K(\sqrt{5}, \sqrt{7}) &&
K_1(\sqrt{7+2\sqrt{5}}) && 2 && D_4 & \cr			% 1235
& 4085 && C_2 && K(\sqrt{5}) && K_1(c_2) && 2 && D_5 & \cr	% 1243
& 4108 && C_2 && K(\sqrt{13}) && K_1(a_4) && 2 && D_3 & \cr	% 1251
& 4173 && C_4 &&
K(\sqrt{\smash{(59+13\sqrt{13})/2}\vphantom{-3+4\sqrt{2}}})
&& K_1(a_5) && 2 && Q_{12} & \cr				% 1271
& 4264 && V_4 && K(\sqrt{2}, \sqrt{13}) &&
K_1(\sqrt{7+2\sqrt{2}}) && 2 && D_4 & \cr			% 1296
& 4305 && C_4\times C_2 &&
K(\sqrt{5}, \sqrt{\smash{(43+7\sqrt{21})/2}\vphantom{-3+4\sqrt{2}}})
&& K_1(\sqrt{\smash{3(13+\sqrt{5})/2}\vphantom{-3+4\sqrt{2}}})
&& 2 && C_4\ltimes C_4 & \cr					% 1309
& 4369 && C_2 && K(\sqrt{17}) && K_1(a_3) && 2 && D_3 & \cr      % 1327
& 4460 && C_2 && K(\sqrt{5}) && K_1(a_{19}) && 2 && D_3 & \cr   % 1356
\hlf
\endtable
\hfil
$$

\newpage

\vcorrection{-.35in}
$$
\table{Continued (under GRH)}
& 4485 && V_4 && K(\sqrt{5},\sqrt{13})
&& K_1(\sqrt{11+2\sqrt{13}}) && 2 && D_4 & \cr   		% 1363
& 4488 && C_4\times C_2 &&
K(\sqrt{2}, \sqrt{\smash{(31+5\sqrt{33})/2}\vphantom{-3+4\sqrt{2}}})
&& K_1(\sqrt{\smash{3(5+2\sqrt{2})}\vphantom{-3+4\sqrt{2}}})
&& 2 && C_4\ltimes C_4 & \cr   					% 1364
& 4505 && V_4 && K(\sqrt{5},\sqrt{17})
&& && 2 && Q_{16} & \cr   					% 1370
& 4520 && V_4 && K(\sqrt{2},\sqrt{5})
&& && 2 && Q_{32} & \cr   					% 1373
& 4524 && V_4 && K(\sqrt{3},\sqrt{13})
&& K_1(\sqrt{9+2\sqrt{13}}) && 2 && D_4 & \cr   		% 1375
& 4616 && C_4 &&
K(\sqrt{33+16\sqrt{2}}) && && 2 && Q_{28} & \cr  		% 1402
& 4620 && C_2^3 &&
K(\sqrt{3}, \sqrt{5}, \sqrt{7}) &&
K_1(\sqrt{\smash{(4\sqrt{3}+5\sqrt{5})(2+\sqrt{5})}\vphantom{-3+4\sqrt{2}}})
&& 2 && Q_8\times C_2 & \cr   					% 1403
& 4641 && V_4\times C_3 &&
K(\sqrt{13}, \sqrt{17}, a_{146}) &&
K_1(\sqrt{\smash{3(9+\sqrt{13})/2}\vphantom{-3+4\sqrt{2}}})
&& 2 && D_4\times C_3 & \cr 					% 1410
& 4745 && V_4 &&
K(\sqrt{5}, \sqrt{13}) &&
K(\sqrt{\smash{(2\sqrt{13}+5\sqrt{5})(2+\sqrt{5})}\vphantom{-3+4\sqrt{2}}})
&& 2 && Q_8 & \cr   						% 1441
& 4785 && V_4 &&
K(\sqrt{5}, \sqrt{29}) && K_1(\sqrt{7+2\sqrt{5}})
&& 2 && D_4 & \cr   						% 1454
& 4809 && C_2 && K(\sqrt{21}) && K_1(a_2) && 2 && D_3 & \cr     % 1461
& 4812 && C_2 && K(\sqrt{3}) && K_1(c_1) && 2 && D_5 & \cr      % 1462
& 4872 && V_4 &&
K(\sqrt{2}, \sqrt{21}) &&
K(\sqrt{\smash{(\sqrt{21}+5\sqrt{29})(5+\sqrt{29})}\vphantom{-3+4\sqrt{2}}})
&& 2 && Q_8 & \cr   						% 1478
& 4920 && V_4 && K(\sqrt{5},\sqrt{6}) &&
K_1(\sqrt{\smash{3(11+4\sqrt{5})}\vphantom{-3+4\sqrt{2}}})
&& 2 && D_4 & \cr   						% 1493
& 4965 && C_2 && K(\sqrt{5}) && K_1(a_{22}) && 2 && D_3 & \cr   % 1506
& 5045 && C_4 &&
K(\sqrt{33+4\sqrt{5}}) && && 2 && Q_{28} & \cr  		% 1529
& 5080 && C_2 && K(\sqrt{5}) && K_1(a_{23}) && 2 && D_3 & \cr   % 1541
& 5213 && C_2 && K(\sqrt{13}) && K_1(c_1) && 2 && D_5 & \cr     % 1582
& 5304 && V_4 && K(\sqrt{6},\sqrt{13}) &&
K_1(\sqrt{\smash{3(9+\sqrt{13})/2}\vphantom{-3+4\sqrt{2}}})
&& 2 && D_4 & \cr   						% 1609
& 5340 && V_4 && K(\sqrt{3},\sqrt{5}) &&
K_1(\sqrt{13+4\sqrt{5}}) && 2 && D_4 & \cr   			% 1619
& 5365 && V_4 && K(\sqrt{5},\sqrt{29}) &&
K_1(\sqrt{7+2\sqrt{5}}) && 2 && D_4 & \cr   			% 1627
& 5372 && C_2 && K(\sqrt{17}) && K_1(a_4) && 2 && D_3 & \cr     % 1630
& 5465 && C_2 && K(\sqrt{5}) && K_1(c_3) && 2 && D_5 & \cr      % 1659
& 5480 && V_4 && K(\sqrt{2},\sqrt{5}) &&
K_1(\sqrt{13+4\sqrt{2}}) && 2 && D_4 & \cr   			% 1664
& 5496 && C_2 && K(\sqrt{6}) && K_1(a_2) && 2 && D_3 & \cr      % 1669
& 5397 && C_2 && K(\sqrt{21}) && K_1(a_3) && 3 && S_4 & \cr     % 1638
& 5505 && C_2 && K(\sqrt{5}) && K_1(a_{25}) && 2 && D_3 & \cr   % 1672
& 5512 && V_4 && K(\sqrt{2},\sqrt{13}) &&
K_1(\sqrt{\smash{(15+\sqrt{13})/2}\vphantom{-3+4\sqrt{2}}})
&& 2 && D_4 & \cr   						% 1674
\hlf
\endtable
\hfil
$$

As a remarkable fact, we note the following.
Except only for the four fields
$\Q(\sqrt{1957}), \Q(\sqrt{2777}), \Q(\sqrt{3981}), \Q(\sqrt{5397})$,
$K_{ur}$ coincides with the Hilbert class field of the genus field of $K$
for $d\leqq 5565$.
The fields $\Q(\sqrt{1957}), \Q(\sqrt{2777}), \Q(\sqrt{3981})$
have an unramified $A_4$-extension which is an $S_4$-extension of $\Q$.
The field $\Q(\sqrt{5397})$ has an unramified $S_4$-extension
which is compositum with an $S_4$-extension of $\Q$.
\enddocument