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

\mag=\magstep1

\document
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 $K$:
$K=K_0\subseteqq K_1\subseteqq K_2\subseteqq\cdots$
($K_{i+1}$ 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 $|d|<1000$
except for $d=-883, -907, -947$, and $-996$.
For $|d|\leqq 467$, $d\ne-427$, the results are unconditional.
We give a table of $G$ for $K$ with $K_{ur}\ne K_1$ 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.

\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 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{(5+2\sqrt{-3})(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}\curlywedge D_6 & \cr				% 134
2 && Q_{16}\lt C_3 & \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					% 142
\hlf
\endtable
$$

\newpage

\vcorrection{-.35in}
$$
\table{Continued (under GRH)}
& 472 && C_6 && K(\sqrt{2},\alpha_{4}) &&
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				% 152
& 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 && Q_{12}\times C_2 & \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				% 172
& 564 && C_4\times C_2 &&
K(\sqrt{-1},\sqrt{1+4\sqrt{3}})
&& K_1(\gamma_1) && 2 && Q_{20}\times C_2 & \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\curlywedge D_3 & \cr			% 177
&&  && 2 && 32\Gamma_3f\lt C_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 && D_3\times C_4 & \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
& 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		% 210
& 695 && C_{24} &&  && K_1(\alpha_{13}) && 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}\curlywedge D_5 & \cr				% 220
&& && 2 && Q_{16}\lt C_5 & \cr
& 728 && C_6\times C_2 &&
K(\sqrt{-7},\sqrt{2},\alpha_{86})
&& && 2 && Q_{16}\times C_3 & \cr				% 225
& 731 && C_{12} && K(\sqrt{-5+2\sqrt{17}},\alpha_{87})
&&  && 3 && \widetilde{A_4}\curlyvee C_8 & \cr			% 226
& 740 && C_8\times C_2 && && && 2 && M_{32} & \cr		% 228
& 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			% 230
& 751 && C_{15} && K(\alpha_{91},\gamma_{39}) &&
K_1(\beta_8) && 3 && \widetilde{A_4}\times C_{5} & \cr		% 231
& 755 && C_{12} && K(\sqrt{13+8\sqrt{5}},\alpha_{92}) &&
 && 2 && Q_{28}\times C_3 & \cr					% 232
\hlf
\endtable
\hfil
$$

\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			% 233
%& 760 && V_4 && K(\sqrt{2},\sqrt{5})
%&& && 2 && Q_{32}\curlywedge D_6 & \cr				% 234
& 760 && V_4 && K(\sqrt{2},\sqrt{5}) && && 2 && Q_{32}\lt C_3 & \cr
& 767 && C_{22} &&  &&
K_1(\alpha_4) && 2 && D_3\times C_{11} & \cr			% 236
& 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	% 244
& 799 && C_{16} &&  && K_1(\gamma_{1}) && 2 && I^{16}_{5} & \cr	% 245
& 820 && C_4\times C_2 &&
K(\sqrt{-1},\sqrt{6+\sqrt{41}}) &&  && 2 && 64\Gamma_8c_2 & \cr % 252
& 824 && C_{20} && K(\sqrt{5+8\sqrt{2}},\gamma_{46})
&& K_1(\gamma_{3}) && 2 && Q_{20}\times C_5 & \cr		% 254
& 835 && C_6 && K(\sqrt{5},\alpha_{102}) &&
&& 2 && D_{11}\times C_3 & \cr					% 257
& 840 && C_2^3 &&
K(\sqrt{-2},\sqrt{-3},\sqrt{5}) &&  && 2 && 32\Gamma_4d & \cr	% 260
& 851 && C_{10} && K(\sqrt{-23},\gamma_{49})
&& K_1(\alpha_1) && 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
& 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
& 915 && C_4\times C_2 &&
K(\sqrt{-3},\sqrt{\smash{(-1+\sqrt{61})/2}\vphantom{3+4\sqrt{2}}})
&&  && 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}\curlywedge D_6)\times C_5 & \cr		% 284
&&  && 2 && (Q_{16}\lt C_3)\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_1) && 2 && I^{16}_3 & \cr	% 289
& 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)\curlywedge D_5 & \cr			% 293
&&  && 2 && (C_4\wr C_2)\lt C_5 & \cr			
& 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) && 3\text{ or }4 && ? & \cr			% 305
\hlf
\endtable
\hfil
$$

Now we explain the notations in our table.
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.
(Here we do not need to consider nonisomorphic fields with same discriminants.)

\newpage

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 $8n$:
$$
\aligned
D_n&=\la\,a,b\,\mid\,a^n=b^2=1,\ b^{-1}ab=a^{-1}\,\ra,\\
Q_{4n}&=\la\,a,b\,\mid\,a^{2n}=1,\ b^2=a^n,\ b^{-1}ab=a^{-1}\,\ra,\\
SD_{8n}&=\la\,a,b\,\mid\,a^{4n}=b^2=1,\ b^{-1}ab=a^{2n-1}\,\ra.
\endaligned
$$

\noindent
$I_n^{2m}$ $(m\geqq 2, n\geqq 3)$ denotes the group of order $2mn$ given by
$$
\la\,a,b\,\mid\,a^{2m}=b^n=1,\ a^{-1}ba=b^{-1}\,\ra.
$$
$M_{2^n}$ $(n\geqq 4)$
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 $A_4$\,:\,$\widetilde{A_4}\cong\SL(2,3)$.

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

Fields with $|d|<1000$, $d\ne -883, -907, -947$
can be classified simply as follows:
\eightpoint
\NoBlackBoxes
$$
\matrix
	\format \l & \l & \l \\
	\matrix
		\format \l \\
		\noalign{\vskip18pt}
		\left\{
		\matrix
		\\
		\noalign{\vskip27pt}
		\\
		\endmatrix
		\right. 
	\endmatrix &
	\hskip-5pt
	\matrix
		\format \l \\
		\noalign{\vskip18pt}
		d_E\nmid d\Bigg\{ \\
		\\
		\noalign{\vskip9pt}
		\\
		d_E\mid d\Bigg\{
	\endmatrix &
	\hskip-3pt
	\matrix
		\format \l & \l & \l \\
		\matrix
			\format \l & \l \\
			\matrix
				\format \l \\
				\noalign{\vskip6pt}
				\hskip-1pt
				d\ne-856,-984,-996\biggm\{ \\
			\endmatrix &
			\hskip-1pt
			\matrix
				\format \l & \l \\
				h(K_g)=\frac{h(K)}{[K_g:K]} \biggm\{ &
				\matrix
					\format \l \\
					h(K)=1 \\
					\noalign{\smallskip}
					h(K)>1
				\endmatrix \\
				h(K_g)>\frac{h(K)}{[K_g:K]} &
			\endmatrix
		\endmatrix &
		\hskip-5pt
		\left.
		\matrix
			\format \l & \c \\
			\cdots & K_{ur}=K \\
			\noalign{\smallskip}
			\cdots & K_{ur}=K_1 \\
			\noalign{\smallskip}
			\cdots & K_{ur}=K_2 \\
		\endmatrix
		\right\} &
		K_{ur}=(K_g)_1 \\
		\noalign{\smallskip}
		\matrix
			\format \l \\
			d=-856,-984,-996 \\
			\noalign{\smallskip}
			\matrix
				\format \l & \l \\
				\hskip-1pt
				d=d_E \Bigm\{ &
				\matrix
					\format \l \\
					d=-p\ \text{\,prime}\\
					\noalign{\smallskip}
					d\ \text{\,composite}
				\endmatrix
			\endmatrix \\
			\noalign{\smallskip}
			d=d'd_E\ 
			(d'\text{\,fundamental quadratic discriminant})
		\endmatrix &
		\hskip-5pt
		\left.
		\matrix
			\format \l & \c \\
			\cdots & l\geqq 3 \\
			\noalign{\smallskip}
			\cdots & K_{ur}=K_3 \\
			\noalign{\smallskip}
			\cdots & l\geqq 2 \\
			\noalign{\smallskip}
			\cdots & l\geqq 3 \\
		\endmatrix
		\right\} &
		K_{ur}\supsetneqq (K_g)_1
	\endmatrix
\endmatrix
$$
\tenpoint
Here, we denote by $K_g$ the genus field of $K$.
We mean by $d_E\mid d$ that the discriminant $d$ of $K$ is
divisible by the discriminant of some quartic number field $E$.
\enddocument