 # 2-Stage Metabelian 3-Groups

## 1.A. All known real quadratic examples:

In the following diagrams, we give the 2-stage metabelian 3-groups G = G(K2|K) that occur for all real quadratic fields K = Q(d1/2) with discriminant 0 < d < 200000 and 3-class group of type (3,3). We denote by G = G1 > G2 > ... > Gi > ... > Gm = 1 the descending central series of class m - 1 of G with Gi+1 = [Gi,G]. In particular, G2 = [G,G] is the commutator subgroup G' of G.

### Types a.3 and a.2

(1) Group of maximal class
(e = 2)
G = G(4,4)(alpha,beta,gamma)
in ZEF 2a(4,4)
with (alpha,beta,gamma) = (0,+-1,0),
resp. (alpha,beta,gamma) = (1,0,0),
of class 3 and of order 34 = 81
 K2 | F3 | K1 / / \ \ N1 N2 N3 N4 \ \ / / K
 G4 = 1 | G3 = (3) | G2 = (3,3) / / \ \ M1 M2 M3 M4 \ \ / / G1 = G
Minimal occurrence:
discriminant d = 32009, resp.
discriminant d = 72329
Principalization type of K in N1,...,N4:
a.3, (0,1,0,0), resp. a.2, (1,0,0,0) [1,3] ,
both designated by a in 

Remarks:
The principalization types (0,1,0,0) and (1,0,0,0) do not determine the associated Galois group G = G(K2|K) uniquely.

Results discovered 2003/02/07:
However, according to top recent computations of the structure of Syl3C(K1) = G(K2|K1) = G2 as (3,3) by Karim Belabas with the aid of PARI, G is determined uniquely for principalization type (1,0,0,0) and uniquely up to the sign of beta for principalization type (0,1,0,0).
Further, exactly the same diagram illustrates the descending central series G = G1 >= G2 >= ... for the real quadratic fields K with discriminants
d = 42817 of principalization type a.3, (0,1,0,0) [1,3] , resp.
d = 94636 of principalization type a.2, (1,0,0,0) [1,3] .

Results discovered 2003/02/27:
In fact, we now have further occurrences of this frequent case:
d = 103809, 114889, 130397, 142097, 151141, 153949, 172252, 173944, 184137, 189237,
according to  and computations of Karim Belabas with the aid of PARI.

For this frequent case, we obviously have generally
Syl3(C(K1)) = (3,3) <==>
G(K2|K) in ZEF 2a(4,4) <==>
family principal factorization type (Delta1,Alpha1,Alpha1,Alpha1) <==>
principalization type either (1,0,0,0) or (0,1,0,0),
i. e., the full 3-class group of K capitulates in
three of the four unramified cyclic cubic extensions N1,...,N4 of K,
in the fourth one, say N1, only a subgroup of type (3) capitulates
and this is either Norm_{N1|K}(C(N1)) for type a.2, (1,0,0,0)
or Norm_{Nj|K}(C(Nj)) with 2 <= j <= 4 for type a.3, (0,1,0,0).

Further, the non-Galois cubic subfields L1,...,L4
of N1,...,N4 have uniformly 3-class number 3.

### Type a.1

(2) Group of maximal class
(e = 2)
G = G(6,6)(alpha,beta,gamma)
in ZEF 2b(6,6)
with (alpha,beta,gamma) =
(0,0,1) resp. (+-1,0,1),
of class 5 and of order 36 = 729
 K2 | F5 | F4 | F3 | K1 / / \ \ N1 N2 N3 N4 \ \ / / K
 G6 = 1 | G5 = (3) | G4 = (3,3) | G3 = (32,3) | G2 = (32,32) / / \ \ M1 M2 M3 M4 \ \ / / G1 = G
Minimal and up to 105 unique occurrence:
discriminant d = 62501
Principalization type of K in N1,...,N4:
a.1, (0,0,0,0) [1,3] ,
designated by a in 

Remarks:
Again, the principalization type (0,0,0,0) does not determine the associated Galois group G = G(K2|K) uniquely.

Result discovered 2003/02/07:
However, according to a top recent computation of the structure of Syl3C(K1) = G(K2|K1) = G2 as (9,9) by Karim Belabas with the aid of PARI, at least beta = 0, the class 5, and the order 729 of G are determined uniquely.

Result discovered 2003/02/27:
In fact, we now have two occurrences of this rare case:
d = 62501 and d = 152949,
according to  and a computation of Karim Belabas with the aid of PARI.
For these fields, I got family principal factorization type (Alpha1,Alpha1,Alpha1,Alpha1)
already in August 1991 at Winnipeg City  .

For this rare case, we obviously have generally
Syl3(C(K1)) = (9,9) <==>
G(K2|K) in ZEF 2b(6,6) <==>
family principal factorization type (Alpha1,Alpha1,Alpha1,Alpha1) <==>
principalization type a.1, (0,0,0,0),
i. e., the full 3-class group of K capitulates in
all four unramified cyclic cubic extensions N1,...,N4 of K.

Further, exactly one of the non-Galois cubic subfields L1,...,L4
of N1,...,N4 has 3-class number 9,
the others have 3-class number only 3.
This fact is due to formulas for the transfers ("Verlagerungen") and
shows that the last parameter of the group G must be gamma = 1.
 References:  Franz-Peter Heider und Bodo Schmithals, Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen, J. reine angew. Math. 336 (1982), 1 - 25  Brigitte Nebelung, Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3,3) und Anwendung auf das Kapitulationsproblem, Inauguraldissertation, Köln, 1989  Daniel C. Mayer, List of discriminants dL<200000 of totally real cubic fields L, arranged according to their multiplicities m and conductors f, 1991, Dept. of Comp. Sci., Univ. of Manitoba

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