2.A. All known cyclic cubic examples:
In the following diagrams,
we give all 2stage metabelian 3groups
G = G(K*_{1}Q)
that occured for cyclic cubic fields K
with 2prime conductors f < 10^{5}
[3].
We denote by
G = G_{1} > G_{2} > ... > G_{i} > ... > G_{m} = 1
the descending central series of class m  1
of G with G_{i+1} = [G_{i},G].
In particular, G_{2} = [G,G] is the commutator subgroup G' of G.
The absolute 3genus field K* is the compositum
K* = k*k~ of k and k~,
the cyclic cubic fields with the two primeconductors dividing f.
Hence K* is a bicyclic bicubic field
that contains yet another cyclic cubic field K~ with conductor f.

Group of maximal class
(e = 2)
G = (3,3)
in ZEF(2,2)
of class 1 and of order 3^{2} = 9

    K*_{1} = K*     
 /   /   \   \  
k   K     K~   k~ 
 \   \   /   /  
    Q     

    G_{2} = 1     
 /   /   \   \  
h   H     H~   h~ 
 \   \   /   /  
    G     

Minimal occurrence:
f = 63 = 3^{2}*7
3class numbers of K and K~:
(3,3)


Group of maximal class
(e = 2)
G = G^{(3)}(alpha,beta,gamma)
in ZEF 1a(3,3)
with (alpha,beta,gamma) = (0,0,0),
of class 2 and of order 3^{3} = 27

    K*_{1}     
         
    K*     
 /   /   \   \  
k   K     K~   k~ 
 \   \   /   /  
    Q     

    G_{3} = 1     
         
    G_{2} = (3)     
 /   /   \   \  
h   H     H~   h~ 
 \   \   /   /  
    G     

Minimal occurrence:
f = 657 = 3^{2}*73
3class numbers of K and K~:
(9,9)


Group of maximal class
(e = 2)
G = G^{(4)}(alpha,beta,gamma)
in ZEF 2a(4,4)
with (alpha,beta,gamma) = (0,1,0),
of class 3 and of order 3^{4} = 81

    K*_{1}     
         
    F_{3}     
         
    K*     
 /   /   \   \  
k   K     K~   k~ 
 \   \   /   /  
    Q     

    G_{4} = 1     
         
    G_{3} = (3)     
         
    G_{2} = (3,3)     
 /   /   \   \  
h   H     H~   h~ 
 \   \   /   /  
    G     

Minimal occurrence:
f = 2439 = 3^{2}*271
3class numbers of K and K~:
(9,9)


Group of second maximal class
(e = 3)
G = G^{(4,5)}((alpha,beta,gamma,delta),rho)
in ZEF 1a(4,5)
with (alpha,beta,gamma,delta) = (0,0,0,0), rho = 0,
of class 3 and of order 3^{5} = 243

    K*_{1}     
         
    F_{3}     
         
    K*     
 /   /   \   \  
k   K     K~   k~ 
 \   \   /   /  
    Q     

    G_{4} = 1     
         
    G_{3} = (3,3)     
         
    G_{2} = (3,3,3)     
 /   /   \   \  
h   H     H~   h~ 
 \   \   /   /  
    G     

Minimal occurrence:
f = 4711 ("Eau de Cologne") = 7*673
3class numbers of K and K~:
(27,27)


Group of second maximal class
(e = 3)
G = G^{(5,6)}((alpha,beta,gamma,delta),rho)
in ZEF 1b(5,6) resp. ZEF 2a(5,6)
with (alpha,beta,gamma,delta) = (0,0,0,0), rho = 1
resp. rho = 0,
of class 4 and of order 3^{6} = 729
with rho != beta  1 resp. rho = 0

    K*_{1}     
         
    F_{4}     
         
    F_{3}     
         
    K*     
 /   /   \   \  
k   K     K~   k~ 
 \   \   /   /  
    Q     

    G_{5} = 1     
         
    G_{4} = (3)     
         
    G_{3} = (3,3,3)     
         
    G_{2} = (3^{2},3,3)     
 /   /   \   \  
h   H     H~   h~ 
 \   \   /   /  
    G     

Minimal occurrence:
f = 5383 = 7*769 resp.
f = 41977 = 13*3229
3class numbers of K and K~:
(27,27) resp. (81,27)


Group of second maximal class
(e = 3)
G = G^{(5,6)}((alpha,beta,gamma,delta),rho)
in ZEF 1b(5,6)
with (alpha,beta,gamma,delta) = (0,0,0,0), rho = 1,
of class 4 and of order 3^{6} = 729
with rho = beta  1 != 0

    K*_{1}     
         
    F_{4}     
         
    F_{3}     
         
    K*     
 /   /   \   \  
k   K     K~   k~ 
 \   \   /   /  
    Q     

    G_{5} = 1     
         
    G_{4} = (3)     
         
    G_{3} = (3,3,3)     
         
    G_{2} = (3,3,3,3)     
 /   /   \   \  
h   H     H~   h~ 
 \   \   /   /  
    G     

Minimal occurrence:
f = 21763 = 7*3109
3class numbers of K and K~:
(27,27)

Remark:
Here, the difference to f = 5383 is that we have
G_{2} = (3,3,3,3) and rho = 1
instead of
G_{2} = (3^{2},3,3) and rho = +1,
which implies different structures of
Syl_{3}C(K*) = G(K*_{1}K*) = G_{2}.

Group of second maximal class
(e = 3)
G = G^{(6,7)}((alpha,beta,gamma,delta),rho)
in ZEF 2b(6,7)
with (alpha,beta,gamma,delta) = (0,0,0,0), rho = +1,
of class 5 and of order 3^{7} = 2187

    K*_{1}     
         
    F_{5}     
         
    F_{4}     
         
    F_{3}     
         
    K*     
 /   /   \   \  
k   K     K~   k~ 
 \   \   /   /  
    Q     

    G_{6} = 1     
         
    G_{5} = (3)     
         
    G_{4} = (3,3)     
         
    G_{3} = (3^{2},3,3)     
         
    G_{2} = (3^{2},3^{2},3)     
 /   /   \   \  
h   H     H~   h~ 
 \   \   /   /  
    G     

Minimal occurrence:
f = 68857 = 37*1861
3class numbers of K and K~:
(81,27)


Group of second maximal class
(e = 3)
G = G^{(7,8)}((alpha,beta,gamma,delta),rho)
in ZEF 2b(7,8)
with (alpha,beta,gamma,delta) = (0,0,0,0), rho = +1,
of class 6 and of order 3^{8} = 6561

    K*_{1}     
         
    F_{6}     
         
    F_{5}     
         
    F_{4}     
         
    F_{3}     
         
    K*     
 /   /   \   \  
k   K     K~   k~ 
 \   \   /   /  
    Q     

    G_{7} = 1     
         
    G_{6} = (3)     
         
    G_{5} = (3,3)     
         
    G_{4} = (3^{2},3)     
         
    G_{3} = (3^{2},3^{2},3)     
         
    G_{2} = (3^{3},3^{2},3)     
 /   /   \   \  
h   H     H~   h~ 
 \   \   /   /  
    G     

Minimal occurrence:
f = 36667 = 37*991
3class numbers of K and K~:
(243,27)


Group of lower than second maximal class
(e = 4)
G = G^{(6,8)}((alpha,beta,gamma,delta),rho)
in ZEF 2a(6,8)
with (alpha,beta,gamma,delta) = (0,0,0,0), rho = 0,
of class 5 and of order 3^{8} = 6561

    K*_{1}     
         
    F_{5}     
         
    F_{4}     
         
    F_{3}     
         
    K*     
 /   /   \   \  
k   K     K~   k~ 
 \   \   /   /  
    Q     

    G_{6} = 1     
         
    G_{5} = (3)     
         
    G_{4} = (3,3,3)     
         
    G_{3} = (3^{2},3,3,3)     
         
    G_{2} = (3^{2},3^{2},3,3)     
 /   /   \   \  
h   H     H~   h~ 
 \   \   /   /  
    G     

Minimal occurrence:
f = 42127 = 103*409
3class numbers of K and K~:
(243,81)


References:
[1] Brigitte Nebelung,
Klassifikation metabelscher 3Gruppen
mit Faktorkommutatorgruppe vom Typ (3,3)
und Anwendung auf das Kapitulationsproblem,
Inauguraldissertation, Köln, 1989
[2] Aïssa Derhem,
Sur les corps cubiques cycliques
de conducteur divisible par deux premiers,
Casablanca, 2002
[3] Daniel C. Mayer,
Class Numbers and Principal Factorizations of Families
of Cyclic Cubic Fields with Discriminant d < 10^{10},
Univ. Graz, Computer Centre, 2002

