\input amstex
\documentstyle{amsppt}
\define\ds{\displaystyle}
\define\Q{{\bold Q}}
\define\PSL{{\operatorname{PSL}}}
\define\SL{{\operatorname{SL}}}
\define\GL{{\operatorname{GL}}}
\define\Cl{{\operatorname{Cl}}}
\define\Gal{{\operatorname{Gal}}}
\define\lt{\ltimes}
\define\rt{\rtimes}
\define\la{\langle}
\define\ra{\rangle}
\NoPageNumbers

\mag=\magstep1

\document

\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 imaginary quadratic number fields $K=\Q(\sqrt{d}),
|d|<1000$ with $K_{ur}\ne K_1$}
\vskip-.5cm
$$
\table{}
& 115 && C_2 && K(\sqrt{5}) && K_1(\alpha_1) && 2 && D_3 & \cr
& 120 && V_4 && K(\sqrt{-3},\sqrt{5}) &&
K_1(\sqrt{\smash{(2\sqrt{2}+\sqrt{5})(2+\sqrt{5})}\vphantom{-3+4\sqrt{2}}})
&& 2 && Q_8 & \cr
& 155 && C_4 &&
K(\sqrt{\smash{(-1+5\sqrt{5})/2}\vphantom{-3+4\sqrt{2}}})
&& K_1(\alpha_2) && 2 && Q_{12} & \cr
& 184 && C_4 && K(\sqrt{-3+4\sqrt{2}}) && K_1(\alpha_1) && 2 && Q_{12} & \cr
& 195 && V_4 && K(\sqrt{-3},\sqrt{5}) && && 2 && Q_{16} & \cr
& 235 && C_2 && K(\sqrt{5}) && K_1(\gamma_1) && 2 && D_5 & \cr
& 248 && C_8 &&  && K_1(\alpha_2) && 2 && I_3^8 & \cr
& 255 && C_6\times C_2 &&
K(\sqrt{5},\root{3}\of{\smash{(9+\sqrt{85})/2}\vphantom{-3+4\sqrt{2}}})
&&
K_1(\sqrt{\smash{(2\sqrt{-3}+\sqrt{5})(2+\sqrt{5})}\vphantom{-3+4\sqrt{2}}})
&& 2 && Q_8\times C_3 & \cr
& 260 && C_4\times C_2 && K(\sqrt{5},\sqrt{8+\sqrt{65}}) 
&& && 2 && M_{16} & \cr
& 276 && C_4\times C_2 && K(\sqrt{-1},\sqrt{13+8\sqrt{3}})
&& K_1(\alpha_1) && 2 && Q_{12}\times C_2 & \cr
& 280 && V_4 && K(\sqrt{-7},\sqrt{5}) && && 2 && Q_{16} & \cr
& 283 && C_3 && K(\alpha_{31}) && K_1(\beta_1) && 3 && \widetilde{A_4} & \cr
& 295 && C_8 &&  && K_1(\alpha_4) && 2 && I_3^8 & \cr
& 299 && C_8 &&  && K_1(\alpha_1) && 2 && I_3^8 & \cr
& 312 && V_4 && K(\sqrt{-3},\sqrt{2}) && && 2 && Q_{16} & \cr
& 331 && C_3 && K(\alpha_{36}) && K_1(\beta_2) && 3 && \widetilde{A_4} & \cr
& 340 && V_4 && K(\sqrt{-1},\sqrt{5}) && && 2 && SD_{16} & \cr
& 355 && C_4 && K(\sqrt{-3+4\sqrt{5}}) && && 2 && Q_{28} & \cr
& 372 && V_4 && K(\sqrt{-1},\sqrt{-3}) && K_1(\alpha_2)
&& 2 && D_6 & \cr
& 376 && C_8 &&  && K_1(\gamma_1) && 2 && I_5^8 & \cr
& 391 && C_{14} &&  && K_1(\alpha_1) && 2 && D_3\times C_7 & \cr
& 395 && C_8 &&  && K_1(\gamma_2) && 2 && I_5^8 & \cr
& 403 && C_2 && K(\sqrt{13}) && K_1(\alpha_2) && 2 && D_3 & \cr
& 408 && V_4 && K(\sqrt{-3},\sqrt{2}) &&
K_1(\sqrt{\smash{-(5+\sqrt{17})/2}\vphantom{-3+4\sqrt{2}}})
&& 2 && D_4 & \cr
& 415 && C_{10} && K(\sqrt{5},\gamma_{18}) &&
K_1(\alpha_6) && 2 && D_3\times C_5 & \cr
& 420 && C_2^3 &&
K(\sqrt{-1},\sqrt{-3},\sqrt{5}) &&  && 2 && 32\Gamma_4c_3 & \cr
& 435 && V_4 && K(\sqrt{-3},\sqrt{5})
&& && 2 && Q_{16}\lt C_3 & \cr
%&& && 2 && Q_{16}\curlywedge D_6 & \cr
& 440 && C_6\times C_2 && K(\sqrt{2},\sqrt{5},\alpha_{50})
&& && 2 && Q_{16}\times C_3 & \cr
& 455 && C_{10}\times C_2 &&
K(\sqrt{-7},\sqrt{5},\gamma_{21}) &&
K_1(\sqrt{\smash{(2\sqrt{13}-3\sqrt{5})(2+\sqrt{5})}\vphantom{-3+4\sqrt{2}}})
&& 2 && Q_8\times C_5 & \cr
\hlf
\endtable
$$

Except for $K=\Q(\sqrt{-427})$,
$\Gal(K_{ur}/K)$ is determined unconditionally for $|d|\leqq 463$.
For $K=\Q(\sqrt{-427})$, $K_1$ is the top of the class field tower,
however, the possibility that $\Gal(K_{ur}/K_1)\cong\PSL(2,7)$
is eliminated under GRH.

\newpage

$$
\table{Continued(under GRH)}
& 472 && C_6 && K(\sqrt{2},\alpha_{54}) &&
K_1(\alpha_4) && 2 && D_3\times C_3 & \cr
& 483 && V_4 && K(\sqrt{-3},\sqrt{-7}) &&
K_1(\alpha_1) && 2 && D_6 & \cr
& 491 && C_9 &&  && K_1(\beta_3) && 3 && \widetilde{A_4}\curlywedge C_9 & \cr
& 515 && C_6 && K(\sqrt{5},\alpha_{60})
&& K_1(\gamma_3) && 2 && D_5\times C_3 & \cr
& 520 && V_4 && K(\sqrt{-2},\sqrt{5}) && && 2 && Q_{24} & \cr
& 527 && C_{18} &&  && K_1(\alpha_2) && 2 && D_3\times C_9 & \cr
& 535 && C_{14} &&  && K_1(\alpha_9) && 2 && D_3\times C_7 & \cr
& 552 && C_4\times C_2 &&
K(\sqrt{-3},\sqrt{-1+2\sqrt{6}})
&& K_1(\alpha_1) && 2 && D_3\times C_4 & \cr
& 555 && V_4 && K(\sqrt{-3},\sqrt{5}) && && 2 && Q_{32} & \cr
& 563 && C_9 &&  && K_1(\beta_4) && 3 && \widetilde{A_4}\curlywedge C_9 & \cr
& 564 && C_4\times C_2 &&
K(\sqrt{-1},\sqrt{1+4\sqrt{3}})
&& K_1(\gamma_1) && 2 && D_5\times C_4 & \cr
& 568 && C_4 &&
K(\sqrt{-1+6\sqrt{2}}) &&  && 2 && Q_{28} & \cr
& 580 && C_4\times C_2 &&
K(\sqrt{5},\sqrt{12+\sqrt{145}})
&&  && 2 && 32\Gamma_3f\lt C_3 & \cr
%&&  && 2 && 32\Gamma_3f\curlywedge D_3 & \cr
& 595 && V_4 && K(\sqrt{-7},\sqrt{5}) && && 2 && Q_{40} & \cr
& 611 && C_{10} && K(\sqrt{13},\gamma_{28}) &&
K_1(\gamma_1) && 2 && D_5\times C_5 & \cr
& 632 && C_{8} &&  && K_1(\gamma_2) && 2 && I_5^8 & \cr
& 635 && C_{10} && K(\sqrt{5},\gamma_{31}) &&
K_1(\gamma_5) && 2 && D_5\times C_5 & \cr
& 643 && C_3 && K(\alpha_{72}) && K_1(\beta_5) && 3 && \widetilde{A_4} & \cr
& 644 && C_8\times C_2 && && K_1(\alpha_1) && 2 && D_3\times C_8 & \cr
& 651 && C_4\times C_2 &&
K(\sqrt{-7},\sqrt{\smash{(13+\sqrt{217})/2}\vphantom{-3+4\sqrt{2}}})
&& K_1(\alpha_2) && 2 && Q_{12}\times C_2 & \cr
& 655 && C_{12} && K(\sqrt{7+6\sqrt{5}},\alpha_{75})
&& K_1(\gamma_6) && 2 && Q_{20}\times C_3 & \cr
& 660 && C_2^3 &&
K(\sqrt{-1},\sqrt{-3},\sqrt{5}) &&  && 2 && 64\Gamma_{15}f_2 & \cr
%& 663 && C_8\times C_2 && && && 2 && 64\Gamma_3p & \cr
& 663 && C_8\times C_2 && && && 2 && \Gamma_{3,2}^2 & \cr
& 664 && C_{10} && K(\sqrt{2},\gamma_{32})
&& K_1(\alpha_6) && 2 && D_3\times C_5 & \cr
& 667 && C_4 &&
K(\sqrt{\smash{(-13+3\sqrt{29})/2}\vphantom{3+4\sqrt{2}}})
&& K_1(\alpha_1) && 2 && Q_{12} & \cr
& 680 && C_6\times C_2 &&
K(\sqrt{-2},\sqrt{5},\alpha_{80}) && && 2 && Q_{16}\times C_3 & \cr
& 687 && C_{12} &&
K(\sqrt{\smash{(11+\sqrt{229})/2}\vphantom{-3+4\sqrt{2}}},%\alpha_{81})
\root{3}\of{\smash{(27+\sqrt{229})/2}\vphantom{-3+4\sqrt{2}}})
&&  && 3 && (S_4\curlywedge C_4)\times C_3 & \cr
& 695 && C_{24} &&  && K_1(\alpha_4) && 2 && I^8_3\times C_3 & \cr
& 696 && C_6\times C_2 &&
K(\sqrt{2},\root{3}\of{99+13\sqrt{58}}) && && 2 && Q_{24}\times C_3 & \cr
& 708 && V_4 && K(\sqrt{-1},\sqrt{-3})
&& K_1(\alpha_4) && 2 && D_6 & \cr
& 715 && V_4 && K(\sqrt{-11},\sqrt{5})
&& && 2 && Q_{16}\lt C_5 & \cr
%&& && 2 && Q_{16}\curlywedge D_5 & \cr
& 728 && C_6\times C_2 &&
K(\sqrt{-7},\sqrt{2},\alpha_{86}) && && 2 && Q_{16}\times C_3 & \cr
& 731 && C_{12} && K(\sqrt{-5+2\sqrt{17}},\alpha_{87})
&&  && 3 && \widetilde{A_4}\curlyvee C_8 & \cr
& 740 && C_8\times C_2 && && && 2 && M_{32} & \cr
& 744 && C_6\times C_2 &&
K(\sqrt{-2},\root{3}\of{63+8\sqrt{62}})
&& K_1(\alpha_2) && 2 && D_3\times C_6 & \cr
& 751 && C_{15} && K(\alpha_{91},\gamma_{39}) &&
K_1(\beta_8) && 3 && \widetilde{A_4}\times C_{5} & \cr
& 755 && C_{12} && K(\sqrt{13+8\sqrt{5}},\alpha_{92}) &&
 && 2 && Q_{28}\times C_3 & \cr
\hlf
\endtable
$$

\newpage

$$
\table{Continued(under GRH)}
& 759 && C_{12}\times C_2 &&
K(\sqrt{-3},\sqrt{\smash{(-5+\sqrt{69})/2}\vphantom{3+4\sqrt{2}}},\alpha_{94})
%K(\sqrt{\smash{(-5+\sqrt{69})/2}\vphantom{-3+4\sqrt{2}}},
%\root{3}\of{\smash{(1861+117\sqrt{253})/2}\vphantom{-3+4\sqrt{2}}})
&& K_1(\alpha_1) && 2 && D_3\times C_{12} & \cr
& 760 && V_4 && K(\sqrt{2},\sqrt{5}) && && 2 && Q_{32}\lt C_3	 & \cr
%& 760 && V_4 && K(\sqrt{2},\sqrt{5}) && && 2 && Q_{32}\curlywedge D_6 & \cr
& 767 && C_{22} &&  && K_1(\alpha_4) && 2 && D_3\times C_{11} & \cr
& 771 && C_6 &&
K(\root{3}\of{16+\sqrt{257}}) && && 3 && S_4\times C_3 & \cr	% 237
& 795 && V_4 && K(\sqrt{-3},\sqrt{5}) && && 2 && Q_{40} & \cr
& 799 && C_{16} &&  && K_1(\gamma_{1}) && 2 && I^{16}_{5} & \cr
& 820 && C_4\times C_2 &&
K(\sqrt{-1},\sqrt{6+\sqrt{41}}) &&  && 2 && \Gamma_{2,3}^1 & \cr
%K(\sqrt{-1},\sqrt{6+\sqrt{41}}) &&  && 2 && 64\Gamma_8c_2 & \cr
& 824 && C_{20} && K(\sqrt{5+8\sqrt{2}},\gamma_{46})
&& K_1(\gamma_{3}) && 2 && Q_{20}\times C_5 & \cr
& 835 && C_6 && K(\sqrt{5},\alpha_{102}) &&
 && 2 && D_{11}\times C_3 & \cr
& 840 && C_2^3 &&
K(\sqrt{-2},\sqrt{-3},\sqrt{5}) &&  && 2 && 32\Gamma_4d & \cr
& 851 && C_{10} && K(\sqrt{-23},\gamma_{49})
&& K_1(\alpha_4) && 2 && D_3\times C_5 & \cr			% 262
& 852 && C_4\times C_2 && K(\sqrt{-1},\sqrt{2+5\sqrt{3}}) &&  &&
2 && Q_{28}\times C_2 & \cr					% 263
& 856 && C_6 && K(\sqrt{2},\alpha_{105}) &&
K_1(\alpha_9) && 4 && \widetilde{S_4}\times C_3 & \cr		% 264
& 868 && C_4\times C_2 &&
K(\sqrt{-1},\sqrt{9+4\sqrt{7}})
&& K_1(\alpha_2) && 2 && Q_{12}\times C_2 & \cr			% 267
%& 883 && C_3 && K(\alpha_{109}) &&  && 1 && ? & \cr		% 271
& 884 && C_8\times C_2 && && && 2 && \Gamma_{3,2}^1 & \cr	% 272
%& 884 && C_8\times C_2 && && && 2 && 64\Gamma_3n_2 & \cr	% 272
& 888 && C_6\times C_2 &&
K(\sqrt{2},\root{3}\of{43+5\sqrt{74}})
&& && 2 && Q_{32}\times C_3 & \cr				% 274
& 895 && C_{16} &&  && K_1(\gamma_{9}) && 2 && I^{16}_5 & \cr	% 275
& 899 && C_{14} &&  && K_1(\alpha_2)
&& 2 && D_3\times C_7 & \cr					% 276
%& 907 && C_3 && K(\alpha_{112}) &&  && 1 && ? & \cr		% 279
& 915 && C_4\times C_2 &&
K(\sqrt{-3},\sqrt{\smash{(-1+\sqrt{61})/2}\vphantom{3+4\sqrt{2}}})
&&  && 2 && \Gamma_{2,3}^2 & \cr				% 281
%&&  && 2 && 64\Gamma_8e & \cr					% 281
& 916 && C_{10} && K(\sqrt{-1},\gamma_{52})
&& && 3 && S_4\times C_5 & \cr					% 282
& 920 && C_{10}\times C_2 &&
K(\sqrt{2},\sqrt{5},\gamma_{53})
&&  && 2 && (Q_{16}\lt C_3)\times C_5 & \cr			% 284
%&&  && 2 && (Q_{16}\curlywedge D_6)\times C_5 & \cr		% 284
& 923 && C_{10} && K(\sqrt{13},\gamma_{54})
&& && 2 && D_7\times C_5 & \cr					% 285
& 935 && C_{14}\times C_2 && &&  && 2 && Q_{16}\times C_7 & \cr	% 287
& 943 && C_{16} &&  && K_1(\alpha_4) && 2 && I^{16}_3 & \cr	% 289
%& 947 && C_5 && K(\gamma_{55}) && && 1 && ? & \cr		% 290
& 948 && C_6\times C_2 &&
K(\sqrt{-1},\root{3}\of{80+9\sqrt{79}})
&& K_1(\gamma_2) && 2 && D_5\times C_6 & \cr			% 291
& 952 && C_4\times C_2 &&
K(\sqrt{-7},\sqrt{-9+10\sqrt{2}})
&&  && 2 && (C_4\wr C_2)\lt C_5 & \cr				% 293
%&&  && 2 && (C_4\wr C_2)\curlywedge D_5 & \cr			% 293
& 955 && C_4 &&
K(\sqrt{\smash{(-9+13\sqrt{5})/2}\vphantom{3+4\sqrt{2}}})
&& && 2 && Q_{52} & \cr						% 294
& 984 && C_6\times C_2 &&
K(\sqrt{2},\root{3}\of{9+\sqrt{82}})
&& && 3 && (Q_{16}\lt C_3^2)\times C_3 & \cr			% 301
& 987 && C_4\times C_2 &&
K(\sqrt{-3},\sqrt{\smash{(-1+3\sqrt{21})/2}\vphantom{3+4\sqrt{2}}})
&& K_1(\gamma_1) && 2 && Q_{20}\times C_2 & \cr			% 302
& 995 && C_8 && &&  && 2 && I_9^8 & \cr				% 304
& 996 && C_6\times C_2 &&
K(\sqrt{-1},\root{3}\of{82+9\sqrt{83}})
&& K_1(\alpha_2) && \geqq 3 && ? & \cr				% 305
\hlf
\endtable
\hfil
$$
\newpage

Supplements.

$$
\vbox{
\offinterlineskip
% preamble
\halign{
\strut$#$ \ & $#$ & $#$ \hfil \cr
%data
%d=-455 & : &
%K_2=
%K_1(\sqrt{\smash{(2\sqrt{13}-3\sqrt{5})(2+\sqrt{5})}
%\vphantom{-3+4\sqrt{2}}}) \cr
d=-520 & : & K_2=
K_1(\sqrt{\smash{(2\sqrt{-2}+\sqrt{5})(2+\sqrt{5})}
\vphantom{-3+4\sqrt{2}}},\alpha_8) \cr
}}
$$

For $d=-883, -907, -947$, $K_1$ is the top of the class field tower,
however, the possibility that $\Gal(K_{ur}/K_1)\cong\PSL(2,7)$
is not still eliminated.
We expect $K_{ur}=K_1.$

For $d=-996$, $K$ has an unramified $S_4\times C_6$-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, $V_4=C_2^2=C_2\times C_2$,
$D_n(n\geqq 3)$ is the dihedral group of order $2n$,
$Q_{4n}(n\geqq 2)$ is the generalized quaternion group of order $4n$,
and $SD_{8n}(n\geqq 2)$ is the semi-dihedral group of order $8n$:
$$
\aligned
D_n&=\la\,a,b\,\mid\,a^n=b^2=1,\ b^{-1}ab=a^{-1}\,\ra,\\
Q_{4n}&=\la\,a,b\,\mid\,a^{2n}=1,\ b^2=a^n,\ b^{-1}ab=a^{-1}\,\ra,\\
SD_{8n}&=\la\,a,b\,\mid\,a^{4n}=b^2=1,\ b^{-1}ab=a^{2n-1}\,\ra.
\endaligned
$$
$I_n^{2m}(m\geqq 2, n\geqq 3)$ denotes the group of order $2mn$ given by
$$
\la\,a,b\,\mid\,a^{2m}=b^n=1,\ a^{-1}ba=b^{-1}\,\ra.
$$
%$SA_{8n}(n\geqq 2)$ denotes the group of order $8n$ given by
%$$
%\la\,a,b\,\mid\,a^{4n}=b^2=1,\ b^{-1}ab=a^{2n+1}\,\ra.
%$$
%For each prime $p$, $M_{p^n}(n\geqq 3(n\geqq 4\text{ if $p=2$}))$
%denotes the modular group of order $p^n$ given by
%$$
%\la\,a,b\,\mid\,a^{p^{n-1}}=b^p=1,\ b^{-1}ab=a^{p^{n-2}+1}\,\ra.
%$$
$M_{2^n}(n\geqq 4)$
denotes the modular group of order $2^n$ given by
$$
\la\,a,b\,\mid\,a^{2^{n-1}}=b^2=1,\ b^{-1}ab=a^{2^{n-2}+1}\,\ra.
$$
$\widetilde{A_4}$ is the double cover of $A_4$\,:\,$\widetilde{A_4}\cong\SL(2,3).$
\newline
$\widetilde{S_4}$ is the double cover of $S_4$ with $\widetilde{S_4}\cong\GL(2,3)$.

\noindent
For some $2$-groups
we use designations given in the table by M. Hall and J. K. Senior.
% \cite{2}.
%We note that T. W. Sag and J. W. Wamsley give
%minimal presentations for all $2$-groups of orders$\leqq 2^6$ \cite{3}.
For simplicity, for some of them we use the following designations.
%used in \cite{1}.
$\Gamma_{m,t}^1$ denotes the group of order $2^{m+t+1}$ given by
$$
\la\,a,b\,\mid\,a^4=b^{2^m}=1,\ c=[a,b],\ a^2=c^{2^{t-1}},\ 
[a,c]=c^2,\ [ab,c]=1\,\ra.
$$
$\Gamma_{m,t}^2$ denotes the group of order $2^{m+t+1}$ given by
$$
\la\,a,b\,\mid\,a^4=b^{2^{m+1}}=1,\ c=[a,b],\ a^2=b^{2^m}=c^{2^{t-1}},\ 
[a,c]=c^2,\ [ab,c]=1\,\ra.
$$
We note that $\Gamma_{3,2}^2=64\Gamma_3p, \Gamma_{2,3}^1=64\Gamma_8c_2,
\Gamma_{3,2}^1=64\Gamma_3n_2, \Gamma_{2,3}^2=64\Gamma_8e,
\Gamma_{2,2}^2=32\Gamma_3e$ $(\cong C_4\wr C_2)$.

\enddocument
\Refs

\ref \no 1 \by E. Benjamin, F. Lemmermeyer, and C. Snyder
\paper Imaginary Quadratic Fields $k$ with Cyclic $\Cl_2(k^1)$
\paperinfo preprint, 1996
\endref

\ref \no 2 \by M. Hall, Jr. and J. K. Senior
\book The groups of order $2^n(n\leq 6)$
\publ The Macmillan Co. \publaddr New York \yr 1964
\moreref MR {\bf 29} \#5889
\endref

\ref \no 3 \by T. W. Sag and J. W. Wamsley
\paper Minimal presentations for groups of order $2^n, n\leqq 6$
\jour J. Austral. Math. Soc. \vol 15 \yr 1973 \pages 461--469
\moreref MR {\bf 49} \#406
\endref

\endRefs


Let $K=\Q(\sqrt{d})$ be an imaginary quadratic number field
with discriminant $d<0$ and
$K_{ur}$ its maximal unramified extension.
We denote by $G$ the Galois group $\Gal(K_{ur}/K)$ and by
$l$ 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 $K_i$),
i.e., $l$ is the smallest number with $K_{l+1}=K_l$.

In the simple expressions of $K_1$ and $K_2$,
$\alpha_i, \beta_i$ and $\gamma_i$ denote
any algebraic numbers generating
the $i$th cubic number field of signature $(1,1)$,
the $i$th quartic number field of signature $(2,1)$
with Galois group isomorphic to $S_4$, and
the $i$th 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.
(We do not need to consider nonisomorphic fields with same discriminants.)

We needed to assume the truth of GRH
to determine the degree of $K_{ur}$ for fields
$K=\Q(\sqrt{d})$ with discriminant $d<-420$.
For the field $K=\Q(\sqrt{-723})$, we have $\Cl(K)\cong C_4$ and
the class field tower of $K$ terminates with
$K_1=K(\sqrt{\smash{(-7+3\sqrt{241})/2}\vphantom{3+4\sqrt{2}}})$.
We cannot get $[K_{ur}:K_1]<60$ from available discriminant bounds.
Thus, in order to determine $K_{ur}$,
we need to judge whether $K_1$ has an unramified nonsolvable Galois extension.


For all exhibited fields, $K_{ur}=K_l$.
We needed to assume the truth of GRH
to determine the degree of $K_{ur}$ for fields
$K=\Q(\sqrt{d})$ with discriminant $d<-420$.

For the field $K=\Q(\sqrt{-723})$, we have $\Cl(K)\cong C_4$ and
the class field tower of $K$ terminates with
$K_1=K(\sqrt{\smash{(-7+3\sqrt{241})/2}\vphantom{3+4\sqrt{2}}})$.
We cannot get $[K_{ur}:K_1]<60$ from available discriminant bounds.
Thus, in order to determine $K_{ur}$,
we need to judge whether $K_1$ has an unramified nonsolvable Galois extension.
(If $K_1$ has such an extension, that extension is an $A_5$-extension of $K_1$
and it is nothing but $K_{ur}$ (under GRH).
We expect that $K_1$ has no nontrivial unramified nextension.)
