\input amstex
\documentstyle{amsppt}

\define\Q{{\bold Q}}

\mag=\magstep1
\topmatter

\title List of known nonnormal CM-fields with class number one
\endtitle

\endtopmatter

\document

After proving the finiteness of imaginary abelian number fields
with class numbers less than given bound,
K. Uchida noted that the same argument shows that
there exist only finitely many CM-fields
with class numbers less than given bound of fixed degree.
A. M. Odlyzko proved that
the degrees of the CM-fields with class numbers less than given bound
satisfying the condtion
\roster
\item"" the maximal totally real subfield is the top of a tower of fields
$F_m\supset\cdots\supset F_1\supset F_0=\Q$
such that each $F_i$ is normal over the preceding one
\endroster
are bounded,
and that if we assume the truth of
Generalized Riemann Hypothesis (GRH) or Artin's conjecture,
then we may drop the condition above.
Recently V. K. Murty proved unconditionally
the finiteness of the CM-fields with solvable normal closure
and class numbers less than given bound.
We list here known nonnormal CM-fields with class number one.

We use the following notations.
Let $K$ be a nonnormal CM-field with class number one
and $K_+$ its maximal totally real subfield.
We denote by $d_K$ and $n$
the discriminant of $K$ and the degree of $K$, respectively.
When the Galois group of the normal closure of $K$ is isomorphic to $G$,
we say $K$ to be of type $G$.
We express $K$ as $K_+(\sqrt{-\delta})$,
where $\delta$ is a totally positive algebraic integer of $K_+$.
$\alpha_i$ denotes an algebraic numbers generating
the $i$th totally real nonnormal cubic number field,
where we consider that the totally real nonnormal cubic number fields
are numbered by discriminants up to isomorphism.
In the list of fields of type $S_4\times C_2$,
we take $\alpha_i$ as a root of a special cubic polynomial
which is given below the list of fields.

Up to now (January 2003), the nonnormal CM-fields with class number one
of degrees $n$ with $n\leqq 6$ have been determined.
Such fields of higher degrees have not been determined yet.
The numbers of (isomorphism) such CM-fields of type $G$ are as follows:
$$
\vbox{
\offinterlineskip
% preamble
\halign{
\strut\vrule#&
\ \hfil $#$ \ & \vrule# & \ \hfil $#$ \ \hfil & \vrule# & \ \hfil $#$ \ &
\vrule# \cr
\noalign{\smallskip}
\noalign{\hrule}
& n && G && \# & \cr
\noalign{\hrule}
&  4 && D_4 && 37 & \cr
&  6 && A_4\times C_2 && 19 & \cr
&  6 && D_6 &&  124 & \cr
&  6 && S_4\times C_2 &&  233 & \cr
\noalign{\hrule}
}}
$$
As usual, $C_n$ is the cyclic group of order $n$,
$D_n$ $(n\geqq 3)$ is the dihedral group of order $2n$,
$A_4$ (resp. $S_4$) is the alternating (resp. symmetric) group of degree four.

\newpage

\NoBlackBoxes

(I) $n=4$.

S. Louboutin and R. Okazaki have determined
the nonnormal quartic CM-fields with class number one.
There exist exactly $37$ nonisomorphic such fields.
All of them are of type $D_4$,
that is, their normal closures have Galois group
isomorphic to $D_4$, the dihedral group of order eight.

\NoBlackBoxes
\def\hlf{\noalign{\hrule}}
\def\vspa{height2pt&\omit&&\omit&&\omit&\omit&&\omit& \cr}
\def\vspb{height1pt&\omit&&\omit&&\omit&\omit&&\omit& \cr}
\def\tableA#1{
\vbox\bgroup\offinterlineskip
\halign\bgroup
 \vrule##&\ $##$ \ \hfil &
 \vrule##&\ $##$ \ \hfil &
% \vrule##&\strut\ $##$ \hfil &
 \vrule##&\strut\ \hfil $##$ & $##$ \hfil &
 \vrule##&\ \hfil $##$ \ &
 \vrule## \cr                            % preamble
\multispan{8} \hfil {#1} \hfil \cr
\noalign{\vskip 0.3cm}
\hlf                                         % full hline
\vspa                                        % wide 2pt
\vspb
& \multispan{1} \hfil {$K$} \hfil &&
\multispan{1} \hfil {$K_+$} \hfil &&
\multispan{1} \hfil {$d_K$} & \hfil &&
rd_K & \cr
\vspb                                        % wide 1pt
\hlf
\vspb
}
\def\endtable{\egroup\egroup}

$$
\tableA{Type $D_4$. ($37$ fields (up to isomorphism))}
& \Q(\sqrt{\smash{-(13+\sqrt{5})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{5}) && 1025&=5^2\cdot 41 &&  5.658 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(5+2\sqrt{2})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{2}) && 1088&=2^6\cdot 17 &&  5.743 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(17+3\sqrt{5})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{5}) && 1525&=5^2\cdot 61 &&  6.249 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(5+\sqrt{17})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{17}) && 2312&=2^3\cdot 17^2 &&  6.934 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(21+\sqrt{5})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{5}) && 2725&=5^2\cdot 109 &&  7.225 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(9+\sqrt{13})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{13}) && 2873&=13^2\cdot 17 &&  7.321 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(13+2\sqrt{5})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{5}) && 3725&=5^2\cdot 149 &&  7.812 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(9+2\sqrt{17})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{17}) && 3757&=17^2\cdot 13 &&  7.829 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(9+2\sqrt{2})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{2}) && 4672&=2^6\cdot 73 &&  8.267 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(9+2\sqrt{13})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{13}) && 4901&=13^2\cdot 29 &&  8.367 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(11+4\sqrt{2})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{2}) && 5696&=2^6\cdot 89 &&  8.687 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(17+2\sqrt{5})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{5}) && 6725&=5^2\cdot 269 &&  9.055 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(13+2\sqrt{41})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{41}) && 8405&=5\cdot 41^2 &&  9.574 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(41+5\sqrt{5})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{5}) && 9725&=5^2\cdot 389 &&  9.930 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(9+\sqrt{29})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{29}) && 10933&=13\cdot 29^2 && 10.225 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(19+8\sqrt{2})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{2}) && 14912&=2^6\cdot 233 && 11.050 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(17+2\sqrt{2})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{2}) && 17984&=2^6\cdot 281 && 11.580 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(17+2\sqrt{61})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{61}) && 18605&=5\cdot 61^2 && 11.679 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(25+6\sqrt{13})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{13}) && 26533&=13^2\cdot 157 && 12.762 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(29+3\sqrt{13})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{13}) && 30589&=13^2\cdot 181 && 13.224 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(35+8\sqrt{17})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{17}) && 39593&=17^2\cdot 137 && 14.106 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(9+\sqrt{73})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{73}) && 42632&=2^3\cdot 73^2 && 14.369 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(13+2\sqrt{29})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{29}) && 44573&=29^2\cdot 53 && 14.530 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(21+2\sqrt{109})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{109}) && 59405&=5\cdot 109^2 && 15.611 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(11+\sqrt{89})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{89}) && 63368&=2^3\cdot 89^2 && 15.866 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(23+4\sqrt{17})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{17}) && 74273&=17^2\cdot 257 && 16.508 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(13+\sqrt{53})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{53}) && 81461&=29\cdot 53^2 && 16.894 & \cr
\vspb\hlf%\vspb
\endtable
$$
\vfil
\newpage

$$
\tableA{Continued}
& \Q(\sqrt{\smash{-(13+\sqrt{149})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{149}) && 111005&=5\cdot 149^2 && 18.253 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(35+\sqrt{137})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{137}) && 319073&=17\cdot 137^2 && 23.778 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(25+\sqrt{157})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{157}) && 320437&=13\cdot 157^2 && 23.792 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(17+\sqrt{269})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{269}) && 361805&=5\cdot 269^2 && 24.525 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(29+2\sqrt{181})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{181}) && 425893&=13\cdot 181^2 && 25.546 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(19+\sqrt{233})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{233}) && 434312&=2^3\cdot 233^2 && 25.671 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(103+12\sqrt{73})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{73}) && 516913&=73^2\cdot 97 && 26.813 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(17+\sqrt{281})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{281}) && 631688&=2^3\cdot 281^2 && 28.191 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(103+\sqrt{97})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{97}) && 686857&=73\cdot 97^2 && 28.788 & \cr
\vspb\hlf\vspb
& \Q(\sqrt{\smash{-(41+2\sqrt{389})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{389}) && 756605&=5\cdot 389^2 && 29.492 & \cr
\hlf
\endtable
$$

There exists exactly one other nonnormal quartic CM-field
with relative class number one.

\def\tableAS{
\vbox\bgroup\offinterlineskip
\halign\bgroup
 \vrule##&\ $##$ \ \hfil &
 \vrule##&\ $##$ \ \hfil &
 \vrule##&\strut\ $##$ & $##$ \hfil &
 \vrule##&\ \hfil $##$ \ &
 \vrule## \cr                            % preamble
\hlf                                         % full hline
\vspa                                        % wide 2pt
\vspb
& \multispan{1} \hfil {$K$} \hfil &&
\multispan{1} \hfil {$K_+$} \hfil &&
\multispan{1} \hfil {$d_K$} & \hfil &&
rd_K & \cr
\vspb                                        % wide 1pt
\hlf
\vspb
}
$$
\tableAS
& K_+(\sqrt{\smash{-(23+2\sqrt{17})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{257}) && 1122833&=17\cdot 257^2 && 32.552 & \cr
\hlf
\endtable
$$

\noindent
For this field, the class number of $K_+$ is three.

These $38$ fields with relative class number one
are divided into $19$ pairs of fields whose normal closures coincide;
Any nonnormal quartic CM-field $K$ with odd relative class number
is of the form $\Q(\sqrt{p}, \sqrt{-\xi})$,
where $p$ is a prime number $\not\equiv 3\pmod{4}$
and $\xi$ is a totally positive generator of a prime ideal
above a prime number $q$ that splits in $\Q(\sqrt{p})$
Then the normal closure $N$ of $K$ is a dihedral octic CM-field that is
a cyclic quartic extension of the real quadratic number field $\Q(\sqrt{pq})$
unramified at all finite primes,
and any one $K_*$ of the two nonnormal quartic subfields of $N$
not isomorphic to $K$ is of the form $\Q(\sqrt{q}, \sqrt{-\eta})$,
where $\eta$ is a totally positive generator of a prime ideal
above a prime number $p$, that splits in $\Q(\sqrt{q})$.
Moreover, the relations $h^-(N)=h^-(K)^2=h^-(K_*)^2$ hold.
Therefore these $19$ pairs correspond one to one to
the dihedral octic CM-fields with relative class number one,
and their simple expressions as pairs $\{p, q\}$ are as follows:
$$
\aligned
&\{2, 17\},\ \{2, 73\},\ \{2, 89\},\ \{2, 233\},\ \{2, 281\},\ 
\{5, 41\},\ \{5, 61\},\ \{5, 109\},\ \{5, 149\},\\
& \{5, 269\},\ \{5, 389\},\ 
\{13, 17\},\ \{13, 29\},\ \{13, 157\},\ \{13, 181\},\ 
\{17, 137\},\ \{17, 257\},\\
&\{29, 53\},\ \{73, 97\}.
\endaligned
$$
\newpage

(II) $n=6$.
\medpagebreak

G. Boutteaux and S. R. Louboutin announced that
G. Boutteaux has completed
the determination of the nonnormal sextic CM-fields with class number one
under supervision by S. R. Louboutin.
There exist exactly $386$ nonisomorphic nonnormal sextic CM-fields
with class number one;
$19$ of them are of type $A_4\times C_2$,
$134$ of them are of type $D_6$, and
the other $233$ are of type $S_4\times C_2$.
\medpagebreak

(i) The case where $K_+$ is normal (cyclic cubic).
\medpagebreak

The nonnormal sextic CM-fields whose maximal totally real subfields are normal
are of type $A_4\times C_2$.
There exist exactly $19$ nonisomorphic nonnormal sextic CM-fields
of type $A_4\times C_2$ with class number one.
Any nonnormal sextic CM-field $K$ of type $A_4\times C_2$
with odd relative class number is of the form $K_+(\sqrt{-\delta})$,
where $K_+$ is a cyclic cubic number field
with odd narrow class number $h_+(K_+)$
and $\alpha$ is a totally positive generator of the $h_+(K_+)$th power
of a prime ideal $\frak{Q}$ in $K_+$ of degree one.
(They call such a field a simplest sextic CM-field.)
The isomorphism class of such a field is determied by
the conductor $f_{K_+}$ of $K_+$ and the rational prime $q$ below $\frak{Q}$.
For each such $K$ with class number one, in addition to $d_K$ and $rd_K$,
we give $f_{K_+}$, $q$ and the minimal polynomial $P(X)$ of some $\alpha$.

\def\vspaa{height2pt&\omit&&\omit&&\omit&&\omit&\omit&&\omit& \cr}
\def\vspbb{height1pt&\omit&&\omit&&\omit&&\omit&\omit&&\omit& \cr}
\def\tableB#1{
\vbox\bgroup\offinterlineskip
\halign\bgroup
 \vrule## & \ \hfil $##$ \ &
 \vrule## & \ \hfil $##$ \ &
 \vrule## & \ $##$ \ \hfil &
 \vrule## & \strut\ \hfil $##$ & $##$ \hfil &
 \vrule## & \ \hfil $##$ \ &
 \vrule## \cr                            % preamble
\multispan{12} \hfil {#1} \hfil \cr
\noalign{\vskip 0.3cm}
\hlf                                         % full hline
\vspaa                                       % wide 2pt
\vspbb
& \multispan{1} \hfil {$f_{K_+}$} &&
\multispan{1} \hfil {$q$} \ &&
\multispan{1} \hfil {$P(X)$} \hfil &&
\multispan{1} \hfil {$d_K$} & \hfil && rd_K & \cr
\vspbb                                       % wide 1pt
\hlf
\vspbb
}

$$
\tableB{Type $A_4\times C_2$. ($19$ fields (up to isomorphism))}
&  7 &&  167 && X^3-18X^2+101X-167 && -400967&=-7^4\cdot 167 &&  8.587 & \cr
%-5,15,-31,45,-45,27
\vspbb\hlf\vspbb
&  9 &&   71 && X^3-30X^2+117X-71  && -465831&=-3^8\cdot 71 &&  8.804 & \cr
%-3,6,-5,6,0,3
\vspbb\hlf\vspbb
&  7 &&  239 && X^3-19X^2+118X-239 && -573839&=-7^4\cdot 239 &&  9.115 & \cr
%-1,4,-3,8,-4,8
\vspbb\hlf\vspbb
&  7 && 251 && X^3-22X^2+145X-251 && -602651&=-7^4\cdot 251 &&  9.190 & \cr
%-1,5,-6,14,-7,7
\vspbb\hlf\vspbb
&  7 &&  379 && X^3-26X^2+181X-379 && -909979&=-7^4\cdot 379 &&  9.844 & \cr
%-1,6,-2,9,3,13
\vspbb\hlf\vspbb
&  7 &&  491 && X^3-26X^2+209X-491 && -1178891&=-7^4\cdot 491 && 10.278 & \cr
%-1,7,2,14,5,13
\vspbb\hlf\vspbb
&  9 &&  199 && X^3-39X^2+318X-199  && -1305639&=-3^8\cdot 199 && 10.454 & \cr
%0,3,-1,6,0,8
\vspbb\hlf\vspbb
&  7 &&  547 && X^3-27X^2+222X-547 && -1313347&=-7^4\cdot 547 && 10.464 & \cr
%-1,6,-2,16,-4,13
\vspbb\hlf\vspbb
& 13 &&   47 && X^3-15X^2+62X-47  && -1342367&=-13^4\cdot 47 && 10.502 & \cr
%-2,4,0,5,-8,5
\vspbb\hlf\vspbb
& 13 &&   79 && X^3-14X^2+61X-79  && -2256319&=-13^4\cdot 79 && 11.452 & \cr
%-2,3,-3,6,-8,8
\vspbb\hlf\vspbb
&  9 &&  379 && X^3-30X^2+237X-379  && -2486619&=-3^8\cdot 379 && 11.639 & \cr
%-3,9,-14,18,-9,17
\vspbb\hlf\vspbb
&  7 && 1051 && X^3-34X^2+341X-1051 && -2523451&=-7^4\cdot 1051 && 11.668 &
\cr %-1,11,4,29,24,29
\vspbb\hlf\vspbb
&  9 &&  523 && X^3-57X^2+507X-523  && -3431403&=-3^8\cdot 523 && 12.281 & \cr
%-3,9,-12,24,-21,19
\vspbb\hlf\vspbb
& 19 &&   31 && X^3-11X^2+34X-31  && -4039951&=-19^4\cdot 31 && 12.620 & \cr
%0,11,0,34,0,31
\vspbb\hlf\vspbb
&  9 &&  739 && X^3-33X^2+315X-739  && -4848579&=-3^8\cdot 739 && 13.009 & \cr
%-3,12,-19,33,-24,17
\vspbb\hlf\vspbb
& 31 &&   31 && X^3-12X^2+17X-2  && -7388168&=-2^3\cdot 31^4 && 13.955 & \cr
%0,26,0,29,0,8
\vspbb\hlf\vspbb
& 19 &&   83 && X^3-18X^2+89X-83  && -10816643&=-19^4\cdot 83 && 14.871 & \cr
%0,18,0,89,0,83
\vspbb\hlf\vspbb
& 37 &&   11 && X^3-10X^2+21X-11  && -20615771&=-11\cdot 37^4 && 16.558 & \cr
\vspbb\hlf\vspbb
& 61 &&    3 && X^3-15X^2+14X-3  && -41537523&=-3\cdot 61^4 && 18.609 & \cr
\hlf
\endtable
$$
\newpage

(ii) The case where $K_+$ is nonnormal.
\medpagebreak

Nonnormal sextic CM-fields
whose maximal totally real subfields are nonnormal
are of type $D_6$ or $S_4\times C_2$
according as they contain an imaginary quadratic number field or do not.

There exist exactly $134$ nonisomorphic nonnormal sextic CM-fields
of type $D_6$ with class number one.
Each nonnormal sextic CM-field $K$ of type $D_6$ is
the compositum of a nonnormal totally real cubic number field $K_+$
with an imaginary quadratic number field $F$
and if $K$ has class number one, then also $F$ has class number one.
For $K$ with class number one,
the isomorphism class of $K_+$ is uniquely determined by its discriminant.
Moreover, in this case only one rational finite prime $q$ is ramified in $F$.
$134$ fields are divided into three
by the factorization type (abbrev. f-type) of $q$ in $K_+$,
$3$ (inert), $1^3$ (totally ramified), or $1^21$ (partially ramified).
($111$ can never occur ($q$ can never split completely in $K_+$).)

\def\tableC#1{
\vbox\bgroup\offinterlineskip
\halign\bgroup
 \vrule## & \ $##$ \ \hfil &
 \vrule## & \hfil \ $##$ \ \hfil &
 \vrule## & \strut\ \hfil $##$ & $##$ \hfil &
 \vrule## &\ \hfil $##$ \ &
 \vrule## \cr                            % preamble
\multispan{9} \hfil {#1} \hfil \cr
\noalign{\vskip 0.3cm}
\hlf                                         % full hline
\vspa                                        % wide 2pt
\vspb
& \multispan{1} \hfil {$K$} \hfil &&
\multispan{1} \hfil {$K_+$} \hfil &&
\multispan{1} \hfil {$d_K$} & \hfil &&
rd_K & \cr
\vspb                                        % wide 1pt
\hlf
\vspb
}
$$
\tableC{Type $D_6$ and f-type $3$ ($5$ fields (up to isomorphism))}
& K_+(\sqrt{-3}) &&
\Q(\alpha_1) && -591408&=-2^4\cdot 3^3\cdot 37^2 &&  9.161 & \cr
%-1,4,1,10,-3,1
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_4) && -2696112&=-2^4\cdot 3^3\cdot 79^2 && 11.797 & \cr
%-1,5,0,18,-8,4
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_5) && -6594624&=-2^6\cdot 3^2\cdot 107^2 && 13.694 & \cr
%2,-4,-6,19,8,29
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{10}) && -8710848&=-2^6\cdot 3^3\cdot 71^2 && 14.344 & \cr
%-1,7,10,34,12,4
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{20}) && -23857200&=-2^4\cdot 3^3\cdot 5^2\cdot 47^2 && 16.966 & \cr
%0,7,-8,49,-28,16
\vspb\hlf%\vspb
}}
$$

$$
\tableC{Type $D_6$ and f-type $1^3$
($26$ fields (up to isomorphism))}
& K_+(\sqrt{-1}) &&
\Q(\alpha_1) && -350464&=-2^8\cdot 37^2 &&  8.396 & \cr %2,-2,-4,12,8,20
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{11}) && -1156923&=-3^7\cdot 23^2 && 10.245 & \cr %-3,0,9,3,-18,12
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{14}) && -1714608&=-2^4\cdot 3^7\cdot 7^2 && 10.940 & \cr
%-3,0,9,6,-9,3
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{18}) && -2101707&=-3^7\cdot 31^2 && 11.317 & \cr %-3,0,13,0,-3,31
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_4) && -2611456&=-2^8\cdot 101^2 && 11.734 & \cr
%-2,-6,4,32,24,36
\vspb\hlf\vspb
& K_+(\sqrt{-2}) &&
\Q(\alpha_1) && -2803712&=-2^{11}\cdot 37^2 && 11.874 & \cr %2,1,0,23,14,59
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_9) && -5089536&=-2^8\cdot 3^2\cdot 47^2 && 13.115 & \cr
%-2,2,2,16,-24,18
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{44}) && -7873200&=-2^4\cdot 3^9\cdot 5^2 && 14.104 & \cr
%0,12,0,36,0,12
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_{14}) && -9144576&=-2^8\cdot 3^6\cdot 7^2 && 14.460 & \cr
%0,12,0,36,0,4
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{53}) && -11337408&=-2^6\cdot 3^{11} && 14.988 & \cr %0,-9,0,81,-54,12
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{64}) && -15066243&=-3^7\cdot 83^2 && 15.715 & \cr %-3,-3,4,42,15,19
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{77}) && -21870000&=-2^4\cdot 3^7\cdot 5^4 && 16.722 & \cr
%-3,-6,7,108,-177,73
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{82}) && -23654592&=-2^6\cdot 3^7\cdot 13^2 && 16.942 & \cr
%0,9,-4,81,-18,4
\vspb\hlf\vspb
& K_+(\sqrt{-11}) &&
\Q(\alpha_{42}) && -27217619&=-11^5\cdot 13^2 && 17.343 & \cr
%-1 5 -6 20 -16 64
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{93}) && -29428272&=-2^4\cdot 3^7\cdot 29^2 && 17.570 & \cr
%-3,-3,18,21,-27,237
\vspb\hlf%\vspb
}}
$$
\vfil
\newpage
$$
\tableC{Continued}
& K_+(\sqrt{-1}) &&
\Q(\alpha_{40}) && -37161216&=-2^8\cdot 3^2\cdot 127^2 && 18.267 & \cr
%0,15,0,51,0,1
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{131}) && -50528448&=-2^6\cdot 3^7\cdot 19^2 && 19.227 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{160}) && -70848800&=-2^4\cdot 3^{11}\cdot 5^2 && 20.342 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{202}) && -105850800&=-2^4\cdot 3^7\cdot 5^5\cdot 11^2 && 21.749 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{226}) && -130205232&=-2^4\cdot 3^7\cdot 611^2 && 22.513 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_{92}) && -156150016&=-2^8\cdot 11^2\cdot 71^2 && 23.205 & \cr
%0,0,-6,64,-48,18
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{298}) && -206122563&=-3^7\cdot 307^2 && 24.304 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{335}) && -251332227&=-3^9\cdot 113^2 && 25.121 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{457}) && -423403200&=-2^6\cdot 3^7\cdot 5^2\cdot 11^2 && 27.402 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{491}) && -487228608&=-2^6\cdot 3^7\cdot 59^2 && 28.051 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{837}) && -1246644675&=-3^7\cdot 5^2\cdot 151^2 && 32.806 & \cr
\vspb\hlf%\vspb
}}
$$

$$
\tableC{Type $D_6$ and f-type $1^21$
($103$ fields (up to isomorphism))}
& K_+(\sqrt{-3}) &&
\Q(\alpha_5) && -309123&=-3^3\cdot 107^2 &&  8.222 & \cr %-3,1,6,1,-6,3
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_4) && -399424&=-2^6\cdot 79^2 &&  8.581 & \cr %2,-4,-8,17,22,34
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_9) && -954288&=-2^4\cdot 3^3\cdot 47^2 &&  9.922 & \cr
%-2,1,0,3,-18,27
\vspb\hlf\vspb
& K_+(\sqrt{-7}) &&
\Q(\alpha_7) && -1539727&=-7^3\cdot 67^2 && 10.745 & \cr %-2,0,3,0,-8,8
\vspb\hlf\vspb
& K_+(\sqrt{-11}) &&
\Q(\alpha_8) && -2461019&=-11^3\cdot 43^2 && 11.619 & \cr %-1,-1,2,15,0,11
\vspb\hlf\vspb
& K_+(\sqrt{-2}) &&
\Q(\alpha_{10}) && -2580992&=-2^9\cdot 71^2 && 11.712 & \cr %-2,1,0,2,-8,8
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{22}) && -2958147&=-3^3\cdot 331^2 && 11.981 & \cr %-3,14,-23,37,-26,7
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_{20}) && -3534400&=-2^6\cdot 5^2\cdot 47^2 && 12.342 & \cr
%-2,2,2,25,-40,32
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{25}) && -3636603&=-3^3\cdot 367^2 && 12.400 & \cr
%-1,-3,10,-12,-16,64
\vspb\hlf\vspb
& K_+(\sqrt{-7}) &&
\Q(\alpha_{14}) && -4000752&=-2^4\cdot 3^6\cdot 7^3 && 12.599 & \cr
%0,0,-2,0,0,8
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{35}) && -6091875&=-3^3\cdot 5^4\cdot 19^2 && 13.514 & \cr
%-3,-1,2,28,-37,13
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{40}) && -6967728&=-2^4\cdot 3^3\cdot 127^2 && 13.820 & \cr
%-1,8,5,50,-7,1
\vspb\hlf\vspb
& K_+(\sqrt{-43}) &&
\Q(\alpha_8) && -9620347&=-11^2\cdot 43^3 && 14.583 & \cr
%-1,1,8,2,-15,13
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_{45}) && -11669056&=-2^6\cdot 7^2\cdot 61^2 && 15.060 & \cr
%-2,7,8,-1,54,58
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_{47}) && -12559936&=-2^6\cdot 443^2 && 15.246 & \cr %0,0,-6,49,-42,18
\vspb\hlf\vspb
& K_+(\sqrt{-2}) &&
\Q(\alpha_{30}) && -13603328&=-2^9\cdot 163^2 && 15.450 & \cr %0,-2,-2,-6,0,27
\vspb\hlf\vspb
& K_+(\sqrt{-67}) &&
\Q(\alpha_7) && -14737387&=-7^2\cdot 67^3 && 15.658 & \cr %-2,4,-2,18,-32,17
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{70}) && -18825075&=-3^3\cdot 5^2\cdot 167^2 && 16.310 & \cr
%-1,-8,2,53,99,181
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_{74}) && -27793984&=-2^6\cdot 659^2 && 17.404 & \cr%0,0,-2,49,-14,2
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_{77}) && -29160000&=-2^6\cdot 3^6\cdot 5^4 && 17.544 & \cr
%0,0,-20,0,0,125
\vspb\hlf%\vspb
}}
$$
\vfil
\newpage

$$
\tableC{Continued.}
& K_+(\sqrt{-3}) &&
\Q(\alpha_{95}) && -29654208&=-2^6\cdot 3^3\cdot 131^2 && 17.593 & \cr
%0,14,0,49,0,12
\vspb\hlf\vspb
& K_+(\sqrt{-2}) &&
\Q(\alpha_{53}) && -30233088&=-2^9\cdot 3^{10} && 17.650 & \cr %0,0,-6,0,0,27
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{99}) && -31726512&=-2^{11}\cdot 3^3\cdot 271^2 && 17.792 & \cr
%-1,10,3,84,-27,9
\vspb\hlf\vspb
& K_+(\sqrt{-7}) &&
\Q(\alpha_{60}) && -33175303&=-7^3\cdot 311^2 && 17.925 & \cr %0,10,0,25,0,7
\vspb\hlf\vspb
& K_+(\sqrt{-7}) &&
\Q(\alpha_{63}) && -34904023&=-7^3\cdot 11^2\cdot 29^2 && 18.077 & \cr
%-3,13,-21,35,-25,11
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{109}) && -37594800&=-2^4\cdot 3^3\cdot 5^2\cdot 59^2 && 18.302 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{111}) && -38363328&=-2^6\cdot 3^3\cdot 149^2 && 18.364 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-19}) &&
\Q(\alpha_{35}) && -38581875&=-3^2\cdot 5^4\cdot 19^3 && 18.382 & \cr
%2,0,4,0,-18,27
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{120}) && -45000387&=-3^3\cdot 1291^2 && 18.859 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-11}) &&
\Q(\alpha_{56}) && -45062336&=-2^6\cdot 11^3\cdot 23^2 && 18.864 & \cr
%-1,-3,-8,60,-85,55
\vspb\hlf\vspb
& K_+(\sqrt{-11}) &&
\Q(\alpha_{59}) && -48556211&=-11^3\cdot 191^2 && 19.100 & \cr
%-3,14,-23,29,-18,4
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_{114}) && -51724864&=-2^6\cdot 29^2\cdot 31^2 && 19.302 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-19}) &&
\Q(\alpha_{63}) && -54849179&=-7^2\cdot 11^3\cdot 29^2 && 19.492 & \cr
%-2,-7,1,34,61,37
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{149}) && -66044592&=-2^4\cdot 3^3\cdot 17^2\cdot 23^2 && 20.105 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{152}) && -68087088&=-2^4\cdot 3^3\cdot 397^2 && 20.207 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_{139}) && -76177984&=-2^6\cdot 1091^2 && 20.589 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_{156}) && -93857344&=-2^6\cdot 7^2\cdot 173^2 && 21.317 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-2}) &&
\Q(\alpha_{118}) && -111661568&=-2^9\cdot 467^2 && 21.944 & \cr
\vspb\hlf%\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_{175}) && -114746944&=-2^6\cdot 13^2\cdot 103^2 && 22.043 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-7}) &&
\Q(\alpha_{132}) && -123068743&=-7^3\cdot 599^2 && 22.302 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{218}) &&
-123649200&=-2^4\cdot 3^3\cdot 5^2\cdot 107^2 && 22.320 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{242}) && -148347072&=-2^6\cdot 3^3\cdot 293^2 && 23.008 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-7}) &&
\Q(\alpha_{146}) &&
-150772167&=-3^2\cdot 7^3\cdot 13^2\cdot 17^2 && 23.070 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{259}) && -167133888&=-2^6\cdot 3^3\cdot 311^2 && 23.469 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-19}) &&
\Q(\alpha_{90}) && -173402379&=-3^2\cdot 19^3\cdot 53^2 && 23.614 & \cr
%-1,3,-2,12,-16,64
\vspb\hlf\vspb
& K_+(\sqrt{-7}) &&
\Q(\alpha_{165}) && -181285447&=-7^3\cdot 727^2 && 23.789 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_{245}) &&
-200732224&=-2^6\cdot 7^2\cdot 11^2\cdot 23^2 && 24.197 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-7}) &&
\Q(\alpha_{177}) &&
-201783127&=-3^2\cdot 7^3\cdot 13^2\cdot 59^2 && 24.218 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{294}) &&
-202705200&=-2^4\cdot 3^3\cdot 5^2\cdot 137^2 && 24.236 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_{250}) && -209902144&=-2^6\cdot 1811^2 && 24.378 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{302}) &&
-215324352&=-2^6\cdot 3^3\cdot 353^2 && 24.482 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_{254}) &&
-219276864&=-2^6\cdot 3^2\cdot 617^2 && 24.556 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-19}) &&
\Q(\alpha_{107}) && -232218304&=-2^6\cdot 19^3\cdot 23^2 && 24.792 & \cr
%-3,9,-30,-6,68,92
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{336}) &&
-253478592&=-2^6\cdot 3^3\cdot 383^2 && 25.156 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-7}) &&
\Q(\alpha_{209}) &&
-150772167&=-3^2\cdot 7^3\cdot 293^2 && 25.344 & \cr
\noalign{\hrule}
}}
$$
\vfil
\newpage
$$
\tableC{Continued.}
& K_+(\sqrt{-163}) &&
\Q(\alpha_{30}) && -277167808&=-2^6\cdot 163^3 && 25.534 & \cr
%0,3,-12,43,-18,36
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_{308}) &&
-292820544&=-2^6\cdot 3^2\cdot 23^2\cdot 31^2 && 25.769 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-7}) &&
\Q(\alpha_{227}) &&
-305012407&=-7^3\cdot 23^2\cdot 41^2 && 25.944 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{373}) &&
-312120000&=-2^6\cdot 3^3\cdot 5^4\cdot 17^2 && 26.044 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-11}) &&
\Q(\alpha_{176}) &&
-316969664&=-2^6\cdot 11^3\cdot 61^2 && 26.111 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-19}) &&
\Q(\alpha_{131}) &&
-320013504&=-2^6\cdot 3^6\cdot 19^3 && 26.153 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{392}) && -339692643&=-3^3\cdot 3547^2 && 26.414 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{399}) &&
-350698032&=-2^4\cdot 3^3\cdot 17^2\cdot 53^2 && 26.555 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_{352}) &&
-374499904&=-2^6\cdot 41^2\cdot 59^2 && 26.847 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-7}) &&
\Q(\alpha_{272}) &&
-411265575&=-3^2\cdot 5^2\cdot 7^3\cdot 73^2 && 27.270 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{469}) &&
-447691968&=-2^6\cdot 3^3\cdot 509^2 && 27.658 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-7}) &&
\Q(\alpha_{292}) &&
-460745383&=-7^3\cdot 19^2\cdot 61^2 && 27.791 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-7}) &&
\Q(\alpha_{305}) &&
-506345175&=-3^{10}\cdot 5^2\cdot 7^3 && 28.231 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-11}) &&
\Q(\alpha_{245}) &&
-552013616&=-2^4\cdot 7^2\cdot 11^3\cdot 23^2 && 28.641 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_{458}) &&
-564917824&=-2^6\cdot 2971^2 && 28.751 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-19}) &&
\Q(\alpha_{183}) &&
-580826979&=-3^2\cdot 19^3\cdot 97^2 && 28.885 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-19}) &&
\Q(\alpha_{188}) &&
-600958144&=-2^6\cdot 19^3\cdot 37^2 && 29.049 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{566}) &&
-632491200&=-2^6\cdot 3^3\cdot 5^2\cdot 11^4 && 29.298 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-2}) &&
\Q(\alpha_{327}) && -636531200&=-2^9\cdot 5^2\cdot 223^2 && 29.329 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-7}) &&
\Q(\alpha_{363}) &&
-686763175&=-5^2\cdot 7^3\cdot 283^2 && 29.703 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-7}) &&
\Q(\alpha_{384}) &&
-750292263&=-3^2\cdot 7^3\cdot 17^2\cdot 29^2 && 30.144 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_{553}) &&
-819791424&=-2^6\cdot 3^2\cdot 1193^2 && 30.592 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{669}) && -835568163&=-3^3\cdot 5563^2 && 30.690 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{675}) &&
-852778800&=-2^4\cdot 3^3\cdot 5^2\cdot 281^2 && 30.794 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_{579}) &&
-89472744&=-2^6\cdot 3739^2 && 31.041 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-2}) &&
\Q(\alpha_{412}) && -973640192&=-2^9\cdot 7^2\cdot 197^2 && 31.482 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{729}) && -993829203&=-3^3\cdot 6067^2 && 31.590 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_{631}) &&
-1029639744&=-2^6\cdot 3^2\cdot 7^2\cdot 191^2 && 31.777 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-2}) &&
\Q(\alpha_{439}) && -1077965312&=-2^9\cdot 1451^2 && 32.020 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{760}) &&
-1085281200&=-2^4\cdot 3^3\cdot 5^2\cdot 317^2 && 32.057 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\alpha_{812}) && -1208775987&=-3^3\cdot 6691^2 && 32.638 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_{743}) &&
-1384435264&=-2^6\cdot 4651^2 && 33.384 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-7}) &&
\Q(\alpha_{556}) &&
-1448497575&=-3^2\cdot 5^2\cdot 7^3\cdot 137^2 && 33.637 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-2}) &&
\Q(\alpha_{552}) && -1635004928&=-2^9\cdot 1787^2 && 34.323 & \cr
\noalign{\hrule}
}}
$$
\vfil
\newpage
$$
\tableC{Continued.}
& K_+(\sqrt{-2}) &&
\Q(\alpha_{572}) && -1724019200&=-2^9\cdot 5^2\cdot 367^2 && 34.627 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\alpha_{879}) &&
-1818851904&=-2^6\cdot 3^2\cdot 1777^2 && 34.938 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-7}) &&
\Q(\alpha_{671}) &&
-1960890183&=-3^2\cdot 7^3\cdot 797^2 && 35.378 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-43}) &&
\Q(\alpha_{249}) &&
-2244005568&=-2^6\cdot 3^2\cdot 7^2\cdot 43^2 && 36.182 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-11}) &&
\Q(\alpha_{629}) &&
-2802288731&=-11^3\cdot 1451^2 && 37.547 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-7}) &&
\Q(\alpha_{848}) &&
-2954679175&=-5^2\cdot 7^3\cdot 587^2 && 37.880 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-11}) &&
\Q(\alpha_{685}) &&
-3222532016&=-2^4\cdot 11^3\cdot 389^2 && 38.432 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-7}) &&
\Q(\alpha_{1048}) &&
-4462584007&=-7^3\cdot 3607^2 && 40.575 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-43}) &&
\Q(\alpha_{421}) &&
-5333409067&=-7^2\cdot 37^2\cdot 43^2 && 41.799 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-2}) &&
\Q(\alpha_{1205}) && -6499824128&=-2^9\cdot 7^2\cdot 509^2 && 43.199 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-7}) &&
\Q(\alpha_{1303}) &&
-6661625607&=-3^2\cdot 7^3\cdot 13^2\cdot 113^2 && 43.377 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-11}) &&
\Q(\alpha_{1125}) &&
-8048132411&=-\cdot 11^3\cdot 2459^2 && 44.766 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-11}) &&
\Q(\alpha_{1633}) &&
-15558805691&=-11^3\cdot 13^2\cdot 263^2 && 49.964 & \cr
\vspb\hlf\vspb
& K_+(\sqrt{-19}) &&
\Q(\alpha_{1705}) &&
-28993925824&=-2^6\cdot 19^3\cdot 257^2 && 55.426 & \cr
\vspb\hlf%\vspb
}}
$$
\vfil
\newpage

\enddocument
\newpage

There exist exactly $233$ nonisomorphic nonnormal sextic CM-fields
of type $S_4\times C_2$ with class number one.
Each nonnormal sextic CM-field $K$ of type $S_4\times C_2$ is
a (totally imaginary) quadratic extension of
a nonnormal totally real cubic number field $K_+$
not containing any imaginary quadratic number field.
For $K$ with class number one,
the isomorphism class of $K_+$ is uniquely determined by its discriminant.
Moreover, in this case only one finite prime of $K_+$ is ramified in $K$.
$134$ fields are divided into three
by the factorization type (abbrev. f-type) of $q$ in $K_+$,
$3$ (inert), $1^3$ (totally ramified), or $1^21$ (partially ramified).
($111$ can never occur ($q$ can never split completely in $K_+$).)

$$
\tableC{Type $S_4\times C_2$. ($K_+$\,:\,nonnormal).}
& K_+(\sqrt{-(15-4\alpha_1-4\alpha_1^2)}) &&
\Q(\alpha_1) && -503792&=-2^4\cdot 23\cdot 37^2 &&  8.920 & \cr %0,21,0,83,0,23
\vspb\hlf\vspb
& K_+(\sqrt{-(-1-4\alpha_1+4\alpha_1^2)}) &&
\Q(\alpha_1) && -679024&=-2^4\cdot 31\cdot 37^2 &&  9.375 & \cr %0,29,0,67,0,31
\vspb\hlf\vspb
& K_+(\sqrt{-(16+8\alpha_6-\alpha_6^2)}) &&
\Q(\alpha_6) && -1142512&=-2^4\cdot 7\cdot 101^2 && 10.224 & \cr %-2,2,-2,4,-8,8
\vspb\hlf\vspb
& K_+(\sqrt{-(3+8\alpha_4+4\alpha_4^2)}) &&
\Q(\alpha_4) && -2296688&=-2^4\cdot 23\cdot 79^2 && 11.486 & \cr %-1,2,-2,4,-4,8
\vspb\hlf\vspb
& K_+(\sqrt{-(18+6\alpha_1-\alpha_1^2)}) &&
\Q(\alpha_1) && -2343728&=-2^4\cdot 37^2\cdot 107 && 11.525 & \cr
%-1,2,1,10,-1,5
\vspb\hlf\vspb
& K_+(\sqrt{-(9+11\alpha_5+3\alpha_5^2)}) &&
\Q(\alpha_5) && -2369943&=-3^2\cdot 23\cdot 107^2 && 11.546 & \cr %-2,1,1,2,-8,8
\vspb\hlf\vspb
& K_+(\sqrt{-(7+10\alpha_7+3\alpha_7^2)}) &&
\Q(\alpha_7) && -2419571&=-7^2\cdot 11\cdot 67^2 && 11.586 & \cr
%-3,10,-15,16,-9,2
\vspb\hlf\vspb
& K_+(\sqrt{-(6+22\alpha_1+11\alpha_1^2)}) &&
\Q(\alpha_1) && -3044656&=-2^4\cdot 37^2\cdot 139 && 12.068 & \cr
%-3,10,-15,20,-13,5
\vspb\hlf\vspb
& K_+(\sqrt{-(-1-5\alpha_5+4\alpha_5^2)}) &&
\Q(\alpha_5) && -3194271&=-3^2\cdot 31\cdot 107^2 && 12.135 & \cr
%-2,3,-5,6,-8,8
\vspb\hlf\vspb
& K_+(\sqrt{-(-9-2\alpha_{25}+3\alpha_{25}^2)}) &&
\Q(\alpha_{25}) && -3636603&=-3^3\cdot 367^2 && 12.400 & \cr %0,8,0,12,0,3
\vspb\hlf\vspb
& K_+(\sqrt{-(8-\alpha_{13}^2)}) &&
\Q(\alpha_{13}) && -4298312&=-2^3\cdot 733^2 && 12.751 & \cr %0,9,0,17,0,8
\vspb\hlf\vspb
& K_+(\sqrt{-(3-\alpha_{18})}) &&
\Q(\alpha_{18}) && -5604552&=-2^3\cdot 3^6\cdot 31^2 && 13.327 & \cr
%0,9,0,21,0,8
\vspb\hlf\vspb
& K_+(\sqrt{-(-4-8\alpha_{32}+3\alpha_{32}^2)}) &&
\Q(\alpha_{32}) && -5655387&=-3\cdot 1373^2 && 13.347 & \cr %-3,4,0,3,14,8
\vspb\hlf\vspb
& K_+(\sqrt{-(-1-5\alpha_{13}+2\alpha_{13}^2)}) &&
\Q(\alpha_{13}) && -5910179&=-11\cdot 733^2 && 13.446 & \cr %-3,2,2,6,-27,27
\vspb\hlf\vspb
& K_+(\sqrt{-(40-\alpha_5-7\alpha_5^2)}) &&
\Q(\alpha_5) && -6079419&=-3^2\cdot 59\cdot 107^2 && 13.509 & \cr
%-3,10,-15,17,-10,3
\vspb\hlf\vspb
& K_+(\sqrt{-(8-\alpha_{22}^2)}) &&
\Q(\alpha_{22}) && -6902343&=-3^2\cdot 7\cdot 331^2 && 13.798 & \cr
%0,11,0,26,0,7
\vspb\hlf\vspb
& K_+(\sqrt{-(-20\alpha_6+7\alpha_6^2)}) &&
\Q(\alpha_6) && -7018288&=-2^4\cdot 43\cdot 101^2 && 13.836 & \cr
%-3,10,-15,18,-11,3
\vspb\hlf\vspb
& K_+(\sqrt{-(20-4\alpha_1-5\alpha_1^2)}) &&
\Q(\alpha_1) && -7250224&=-2^4\cdot 37^2\cdot 331 && 13.912 & \cr
%-1,6,-3,6,3,17
\vspb\hlf\vspb
& K_+(\sqrt{-(-5-8\alpha_{11}+4\alpha_{11}^2)}) &&
\Q(\alpha_{11}) && -7327179&=-3^6\cdot 19\cdot 23^2 && 13.936 & \cr
%-3,12,-19,18,-9,2
\vspb\hlf\vspb
& K_+(\sqrt{-(1+3\alpha_5+3\alpha_5^2)}) &&
\Q(\alpha_5) && -8140239&=-3^2\cdot 79\cdot 107^2 && 14.183 & \cr
%-1,2,-3,4,-4,8
\vspb\hlf\vspb
& K_+(\sqrt{-(2+6\alpha_1+7\alpha_1^2)}) &&
\Q(\alpha_1) && -10229168&=-2^4\cdot 37^2\cdot 467 && 14.733 & \cr
%-1,4,-5,12,-9,27
\vspb\hlf\vspb
& K_+(\sqrt{-(18+4\alpha_{10}-3\alpha_{10}^2)}) &&
\Q(\alpha_{10}) && -10323968&=-2^{11}\cdot 71^2 && 14.756 & \cr
%0,7,0,10,0,2
\vspb\hlf\vspb
& K_+(\sqrt{-(11+2\alpha_7-11\alpha_7^2)}) &&
\Q(\alpha_7) && -10338167&=-7^2\cdot 47\cdot 67^2 && 14.759 & \cr %-2,4,1,5,11,7
\vspb\hlf\vspb
& K_+(\sqrt{-(-1-4\alpha_1+8\alpha_1^2)}) &&
\Q(\alpha_1) && -10754864&=-2^4\cdot 37^2\cdot 491 && 14.857 & \cr
%-3,8,-9,18,-5,17
\vspb\hlf\vspb
& K_+(\sqrt{-(3+4\alpha_1+4\alpha_1^2)}) &&
\Q(\alpha_1) && -10930096&=-2^4\cdot 37^2\cdot 499 && 14.897 & \cr
%-3,12,-19,32,-23,13
\vspb\hlf\vspb
& K_+(\sqrt{-(-8\alpha_6+3\alpha_6^2)}) &&
\Q(\alpha_6) && -10935472&=-2^4\cdot 67\cdot 101^2 && 14.898 & \cr
%-1,4,-13,14,-3,7
\vspb\hlf\vspb
& K_+(\sqrt{-(24-8\alpha_1-5\alpha_1^2)}) &&
\Q(\alpha_1) && -11981488&=-2^4\cdot 37^2\cdot 547 && 15.126 & \cr
%-3,12,-29,52,-31,23
\vspb\hlf\vspb
& K_+(\sqrt{-(8-\alpha_{31}^2)}) &&
\Q(\alpha_{31}) && -12663175&=-5^2\cdot 7\cdot 269^2 && 15.267 & \cr
%0,10,0,17,0,7
\vspb\hlf\vspb
& K_+(\sqrt{-(1+2\alpha_4^2)}) &&
\Q(\alpha_4) && -13081136&=-2^4\cdot 79^2\cdot 131 && 15.349 & \cr
%-1,2,-7,6,-9,27
\vspb\hlf\vspb
& K_+(\sqrt{-(7+5\alpha_7+\alpha_7^2)}) &&
\Q(\alpha_7) && -14077504&=-2^6\cdot 7^2\cdot 67^2 && 15.538 & \cr
%0,9,0,14,0,4
\vspb\hlf\vspb
& K_+(\sqrt{-(8-\alpha_{16}^2)}) &&
\Q(\alpha_{16}) && -14173175&=-5^2\cdot 23\cdot 157^2 && 15.556 & \cr
%0,11,0,30,0,23
\vspb\hlf\vspb
& K_+(\sqrt{-(55-12\alpha_1-16\alpha_1^2)}) &&
\Q(\alpha_1) && -15135664&=-2^4\cdot 37^2\cdot 691 && 15.727 & \cr
%-1,8,5,20,7,19
\hlf
\endtable
$$

\newpage

$$
\vbox{
\offinterlineskip
% preamble
\halign{
\strut\vrule#&
\ \hfil $#$ \ & \vrule# & \ \hfil $#$ \ & \vrule# & \ $#$ \hfil \ & \vrule#
\cr
%data
%\multispan{9}{ \hfil Table 6 \hfil }
%\cr
\noalign{\smallskip}
\noalign{\hrule}
& i && d_{K_+} && \text{min. polyn. of $\alpha_i$} & \cr
\noalign{\hrule}
&  1 &&  148 && X^3+X^2-3X-1 & \cr
&  4 &&  316 && X^3+X^2-4X-2 & \cr
&  5 &&  321 && X^3+X^2-4X-1 & \cr
&  6 &&  404 && X^3-X^2-5X-1 & \cr
&  7 &&  469 && X^3+X^2-5X-4 & \cr
& 10 &&  568 && X^3-X^2-6X-2 & \cr
& 11 &&  621 && X^3-6X-3 & \cr
& 13 &&  733 && X^3+X^2-7X-8 & \cr
& 16 &&  785 && X^3+X^2-6X-5 & \cr
& 18 &&  837 && X^3-6X-1 & \cr
& 22 &&  993 && X^3+X^2-6X-3 & \cr
& 25 && 1101 && X^3+X^2-9X-12 & \cr
& 31 && 1345 && X^3-7X-1 & \cr
& 32 && 1373 && X^3-8X-5 & \cr
\noalign{\hrule}
}}
$$
\enddocument
\vbox\bgroup\offinterlineskip
\halign\bgroup
 \vrule##&\ $##$ \ \hfil &
 \vrule##&\ $##$ \ \hfil &
 \vrule##&\strut\ \hfil $##$ & $##$ \hfil &
 \vrule## \cr                            % preamble
\multispan{8} \hfil {#1} \hfil \cr
\noalign{\vskip 0.3cm}
\hlf                                         % full hline
\vspa                                        % wide 2pt
%height0pt & \hphantom{299} && \hphantom{59} &&
%\omit\ \hphantom{1200} &&\hphantom{100} &&\hphantom{ 324720}&
%\hphantom{=2^4 \cdot 3^2 \cdot 5 \cdot 11 \cdot 41} & \cr
\vspb
& \multispan{1} \hfil {$K$} \hfil &&
\multispan{1} \hfil {$K_+$} \hfil && d_K &  & \cr
\vspb                                        % wide 1pt
\hlf
\vspb
}
$$
\tableM{Type $A_4\times C_2$. ($19$ fields (up to isomorphism).}
%& \Q(\sqrt{-(3-4\cos^4(2\pi/7))}) &&
& \Q(\sqrt{-(3-4\cos^4\frac{2\pi}{7})}) &&
%\Q(\cos(2\pi/7)) && -400967&=-7^4\cdot 167 & \cr %-5,15,-31,45,-45,27
\Q(\cos\frac{2\pi}{7}) && -400967&=-7^4\cdot 167 & \cr %-1,5,3,7,7,7
\vspb\hlf\vspb
& \Q(\sqrt{-(7-8\cos\frac{2\pi}{9})}) &&
\Q(\cos\frac{2\pi}{9}) && -465831&=-3^8\cdot 71 & \cr %-3,6,-5,6,0,3
\vspb\hlf\vspb
& \Q(\sqrt{-(14-10\cos\frac{2\pi}{7}-16\cos^2\frac{2\pi}{7})}) &&
\Q(\cos\frac{2\pi}{7}) && -573839&=-7^4\cdot 239 & \cr %-1,4,-3,8,-4,8
\vspb\hlf\vspb
& \Q(\sqrt{-(2-10\cos\frac{2\pi}{7}+11\cos^2\frac{2\pi}{7})}) &&
\Q(\cos\frac{2\pi}{7}) && -602651&=-7^4\cdot 251 & \cr %-1,5,-6,14,-7,7
\vspb\hlf\vspb
& \Q(\sqrt{-(2+22\cos\frac{2\pi}{7}+64\cos^2\frac{2\pi}{7})}) &&
\Q(\cos\frac{2\pi}{7}) && -909979&=-7^4\cdot 379 & \cr %-1,6,-2,9,3,13
\vspb\hlf\vspb
& \Q(\sqrt{-(1-2\cos\frac{2\pi}{7}+3\cos^2\frac{2\pi}{7})}) &&
\Q(\cos\frac{2\pi}{7}) && -1178891&=-7^4\cdot 491 & \cr %-1,7,2,14,5,13
\vspb\hlf\vspb
& \Q(\sqrt{-(1+6\cos\frac{2\pi}{9}+24\cos^2\frac{2\pi}{9})}) &&
\Q(\cos\frac{2\pi}{9}) && -1305639&=-3^8\cdot 199 & \cr %0,3,-1,6,0,8
\vspb\hlf\vspb
& \Q(\sqrt{-(38-2\cos\frac{2\pi}{7}-48\cos^2\frac{2\pi}{7})}) &&
\Q(\cos\frac{2\pi}{7}) && -1313347&=-7^4\cdot 547 & \cr %-1,6,-2,16,-4,13
\vspb\hlf\vspb
& \Q(\sqrt{-(23-6\alpha-\alpha^2)}) &&
K_{13,3}=\Q(\alpha) && -1342367&=-13^4\cdot 47 & \cr %-2,4,0,5,-8,5
\vspb\hlf\vspb
& \Q(\sqrt{-(-4-16\alpha+7\alpha^2)}) &&
K_{13,3}=\Q(\alpha) && -2256319&=-13^4\cdot 79 & \cr %-2,3,-3,6,-8,8
\vspb\hlf\vspb
& \Q(\sqrt{-(19+24\cos\frac{2\pi}{9})}) &&
\Q(\cos\frac{2\pi}{9}) && -2486619&=-3^8\cdot 379 & \cr %-3,9,-14,18,-9,17
\vspb\hlf\vspb
& \Q(\sqrt{-(69-22\cos\frac{2\pi}{7}-108\cos^2\frac{2\pi}{7})}) &&
\Q(\cos\frac{2\pi}{7}) && -2523451&=-7^4\cdot 1051 & \cr %-1,11,4,29,24,29
\vspb\hlf\vspb
& \Q(\sqrt{-(7+38\cos\frac{2\pi}{9}+44\cos^2\frac{2\pi}{9})}) &&
\Q(\cos\frac{2\pi}{9}) && -3431403&=-3^8\cdot 523 & \cr %-3,9,-12,24,-21,19
\vspb\hlf\vspb
& \Q(\sqrt{-(8-\beta^2)}) &&
K_{19,3}=\Q(\beta) && -4039951&=-19^4\cdot 31 & \cr %0,11,0,34,0,31
\vspb\hlf\vspb
& \Q(\sqrt{-(1+4\cos\frac{2\pi}{9}+7\cos^2\frac{2\pi}{9})}) &&
\Q(\cos\frac{2\pi}{9}) && -4848579&=-3^8\cdot 739 & \cr %-3,12,-19,33,-24,17
\vspb\hlf\vspb
& \Q(\sqrt{-(-3-5\gamma+3\gamma^2)}) &&
K_{31,3}=\Q(\gamma) && -7388168&=-2^3\cdot 31^4 & \cr %0,26,0,29,0,8
\vspb\hlf\vspb
& \Q(\sqrt{-(11+2\beta-\beta^2)}) &&
K_{19,3}=\Q(\beta) && -10816643&=-19^4\cdot 83 & \cr %0,18,0,89,0,83
\hlf
\endtable
$$

\enddocument
(We note that such CM-fields are divided into three types;
$A_4\times C_2, D_6$, and $S_4\times C_2$,
where $S_4$ (resp. $A_4$) denotes
the symmetric (resp. alternating) group of degree four and
$C_n$ denotes the cyclic group of order $n$.
The fields of type $A_4\times C_2$ are
CM-extensions of (real) cyclic cubic number fields.
The fields of the other two types are
CM-extensions of nonnormal totally real cubic number fields.
Each of the fields of types $A_4\times C_2$ and $S_4\times C_2$
is of the form $F(\sqrt{-\delta})$,
where $F$ is the (maximal total real) cubic subfield and
$\delta$ is a totally positive irrational element of $F$.)
Therefore, we give also lists for fields of types $D_6$ and $S_4\times C_2$,
but we do not assure that they are complete.
\enddocument
\newpage
For the primes $p=13, 19,$ and $31$,
we denote by $K_{p,3}$ the cubic subfield of
the $p$th cyclotomic number field $\Q(\zeta_p)$.
We denote by $\alpha, \beta$ and $\gamma$ as roots of the polynomials
$X^3-X^2-4X-1, X^3+X^2-6X-7,$ and $X^3+X^2-10X-8$,
which generate $K_{13,3}, K_{19,3},$ and $K_{31,3}$, respectively.
