|
Historical Introduction
In the present investigation, we are re-entering uncharted waters.
In 2002, Aïssa Derhem and ourselves analyzed the relations between
the absolute Galois group g = Gal(F31(K*)|Q)
of the Hilbert 3-class field of the absolute 3-genus field K* = (K|Q)*
of cyclic cubic fields K whose conductor f is divisible by two primes (t = 2)
and the norms of primitive ambiguous principal ideals of K.
First we revisited the scenario of
Ayadi's Thesis
,
where both fields
of the doublet sharing the common conductor f have 3-class number h = 9
and 3-class group of type (3,3). In this case, the metabelian 3-group g turned out to be
of maximal class (coclass 1)
.
Then, however, we pushed forward to the rare cases, where the members
of the doublet have class numbers divisible by 27 and 3-class group
of type (3,9) or (9,9) or (9,27). In contrast to exact 3-class number 9,
the members may have different 3-class numbers, as we observed in our
primary construction process
using Voronoi's algorithm and the Euler product for class number computation.
Then we proved that, for 27 | h, the Galois groups g are
of second maximal class (coclass 2)
,
and in the particularly hard boiled case f = 42127 even
of coclass 3
.
Since this required the computation of the class number of the bicubic field K*,
we gratefully acknowledge that Karim Belabas aided us by means of PARI/GP.
The New Challenge
Now, in the year of our mission Research Frontier 2013, our intention is
to use MAGMA for a break through which would have been impossible in 2002.
Instead of g we are going to determine the relative Galois group
G = Gal(F32(K)|K) of the second Hilbert 3-class field of K
and to show that there are connections between the metabelian 3-groups g and G.
§ 1. Possible second 3-class groups, G = G32(K)
It's a lucky coincidence that we've just started an extensive search for
finite p-groups which may occur as higher p-class groups Gpm(K)
of various types of algebraic number fields K, for m ≥ 2.
Thus we are able to give a list of metabelian 3-groups G having G/G' ≅ (3,9)
referring to our latest view of
coclass trees
for the adopted notation.
|
G
|
G/G'
|
cl
|
cc
|
TKT
|
Mi
|
TTT
|
G'/G''
|
dl
|
Aut
|
σ
|
Rank
|
Desc
|
|
<33,2>
|
(3,9)
|
1
|
2
|
(0000)
|
<32,1>3,<32,2>
|
(9)3,(3,3)
|
1
|
1
|
2233
|
1
|
1/3
|
2/1
|
|
<34,3>
|
(3,9)
|
2
|
2
|
(0000)
|
<33,2>3,<33,5>
|
(3,9)3,(3,3,3)
|
(3)
|
2
|
2235
|
1
|
3/4
|
9/5;18/16;4/4
|
|
<34,6>
|
(3,9)
|
2
|
2
|
(1111)
|
<33,1>3,<33,2>
|
(27)3,(3,9)
|
(3)
|
2
|
2134
|
0
|
0/2
|
0/0
|
|
<35,13>
|
(3,9)
|
3
|
2
|
(0004)
|
<34,3>3,<34,15>
|
(3,9)3,(3,3,3,3)
|
(3,3)
|
2
|
2237
|
1
|
1/4
|
7/4
|
|
<35,14>
|
(3,9)
|
3
|
2
|
(0004)
|
<34,3>3,<34,11>
|
(3,9)3,(3,3,9)
|
(3,3)
|
2
|
2237
|
1
|
1/4
|
6/2
|
|
<35,15>
|
(3,9)
|
3
|
2
|
(0000)
|
<34,3>3,<34,11>
|
(3,9)3,(3,3,9)
|
(3,3)
|
2
|
2237
|
1
|
1/4
|
6/2
|
|
<35,16>
|
(3,9)
|
3
|
2
|
(0003)
|
<34,6>3,<34,11>
|
(3,9)3,(3,3,9)
|
(3,3)
|
2
|
2136
|
1
|
0/3
|
0/0
|
|
<35,17>
|
(3,9)
|
3
|
2
|
(0000)
|
<34,3>2,<34,11>,<34,12>
|
(3,9)2,(3,3,9),(3,3,3)
|
(3,3)
|
2
|
2236
|
1
|
1/4
|
7/2
|
|
<35,18>
|
(3,9)
|
3
|
2
|
(0040)
|
<34,3>2,<34,11>,<34,13>
|
(3,9)2,(3,3,9),(3,3,3)
|
(3,3)
|
2
|
2136
|
1
|
0/3
|
0/0
|
|
<35,19>
|
(3,9)
|
3
|
2
|
(0010)
|
<34,6>2,<34,5>,<34,14>
|
(3,9)2,(3,27),(3,3,3)
|
(3,3)
|
2
|
2235
|
1
|
0/3
|
0/0
|
|
<35,20>
|
(3,9)
|
3
|
2
|
(0010)
|
<34,6>2,<34,5>,<34,14>
|
(3,9)2,(3,27),(3,3,3)
|
(3,3)
|
2
|
2235
|
1
|
0/3
|
0/0
|
|
<36,9>
|
(3,9)
|
3
|
3
|
(0004)
|
<35,32>3,<35,62>
|
(3,3,9)3,(3,3,3,3)
|
(3,3,3)
|
2
|
2239
|
1
|
3/5
|
15/10;61/61;37/37
|
|
<36,10>
|
(3,9)
|
3
|
3
|
(0444)
|
<35,32>4
|
(3,3,9)4
|
(3,3,3)
|
2
|
2238
|
1
|
2/4
|
12/8;13/13
|
|
<36,11>
|
(3,9)
|
3
|
3
|
(0440)
|
<35,32>4
|
(3,3,9)4
|
(3,3,3)
|
2
|
2238
|
1
|
2/4
|
12/8;13/13
|
|
<36,12>
|
(3,9)
|
3
|
3
|
(4444)
|
<35,32>3,<35,63>
|
(3,3,9)3,(3,3,3,3)
|
(3,3,3)
|
2
|
2139
|
1
|
2/4
|
6/4;9/9
|
|
<36,13>
|
(3,9)
|
3
|
3
|
(0113)
|
<35,49>,<35,12>2,<35,35>
|
(3,3,9),(3,27)2,(3,3,9)
|
(3,3,3)
|
2
|
2137
|
1
|
1/3
|
5/5
|
|
<36,14>
|
(3,9)
|
3
|
3
|
(4233)
|
<35,49>,<35,12>2,<35,36>
|
(3,3,9),(3,27)2,(3,3,9)
|
(3,3,3)
|
2
|
2137
|
1
|
0/2
|
0/0
|
|
<36,15>
|
(3,9)
|
3
|
3
|
(4323)
|
<35,49>,<35,12>2,<35,36>
|
(3,3,9),(3,27)2,(3,3,9)
|
(3,3,3)
|
2
|
2137
|
1
|
0/2
|
0/0
|
|
<36,16>
|
(3,9)
|
3
|
3
|
(1114)
|
<35,12>3,<35,64>
|
(3,27)3,(3,3,3,3)
|
(3,3,3)
|
2
|
2238
|
1
|
1/3
|
4/4
|
|
<36,17>
|
(3,9)
|
3
|
3
|
(1234)
|
<35,12>3,<35,34>
|
(3,27)3,(3,3,9)
|
(3,3,3)
|
2
|
2238
|
1
|
1/3
|
3/3
|
|
<36,18>
|
(3,9)
|
3
|
3
|
(1320)
|
<35,12>3,<35,34>
|
(3,27)3,(3,3,9)
|
(3,3,3)
|
2
|
2238
|
1
|
1/3
|
3/3
|
|
<36,19>
|
(3,9)
|
3
|
3
|
(1114)
|
<35,12>3,<35,64>
|
(3,27)3,(3,3,3,3)
|
(3,3,3)
|
2
|
2238
|
1
|
1/3
|
4/4
|
|
<36,20>
|
(3,9)
|
3
|
3
|
(1324)
|
<35,12>3,<35,34>
|
(3,27)3,(3,3,9)
|
(3,3,3)
|
2
|
2238
|
1
|
1/3
|
3/3
|
|
<36,21>
|
(3,9)
|
3
|
3
|
(1230)
|
<35,12>3,<35,34>
|
(3,27)3,(3,3,9)
|
(3,3,3)
|
2
|
2238
|
1
|
1/3
|
3/3
|
In the sequel, we shall encounter exactly the same 55 conductors f < 105
of cyclic cubic fields K with 3-class group Cl3(K) of type (3,9)
which we know from our different investigations in 2002.
Only the particularly extravagant conductor f = 42127 is missing,
since both fields of the corresponding doublet have class number divisible by 81.
There are no conductors divisible by three primes (t = 3) such that
one of the associated four cyclic cubic fields is of type (3,9).
The conductors f in the following Sections § 2.1 and § 2.2
completely contain those conductors f = p*q (t = 2) which are given in
Table 3 on p. 41
of the Section "Tables Numériques"
in the paper by
George Gras
.
§ 2.1. The 13 conductors f divisible by nine, 9 | f
Here, the members of a doublet have equal 3-class numbers 27
and they also have the same TKT a.1, κ(K) = (0000),
and the same 3-class field tower group G32(K) ≅ <81,3>
which we indicate by exponents denoting iteration.
|
No.
|
f
|
factors
|
TKT
|
κ(K)
|
G32(K)
|
|
8
|
20421
|
32*2269
|
a.12
|
(0000)2
|
<81,3>2
|
|
15
|
35163
|
32*3907
|
a.12
|
(0000)2
|
<81,3>2
|
|
24
|
44397
|
32*4933
|
a.12
|
(0000)2
|
<81,3>2
|
|
25
|
49743
|
32*5527
|
a.12
|
(0000)2
|
<81,3>2
|
|
30
|
56223
|
32*6247
|
a.12
|
(0000)2
|
<81,3>2
|
|
32
|
58329
|
32*6481
|
a.12
|
(0000)2
|
<81,3>2
|
|
36
|
64971
|
32*7219
|
a.12
|
(0000)2
|
<81,3>2
|
|
40
|
69183
|
32*7687
|
a.12
|
(0000)2
|
<81,3>2
|
|
42
|
72099
|
32*8011
|
a.12
|
(0000)2
|
<81,3>2
|
|
46
|
78093
|
32*8677
|
a.12
|
(0000)2
|
<81,3>2
|
|
47
|
81009
|
32*9001
|
a.12
|
(0000)2
|
<81,3>2
|
|
49
|
89109
|
32*9901
|
a.12
|
(0000)2
|
<81,3>2
|
|
52
|
95913
|
32*10657
|
a.12
|
(0000)2
|
<81,3>2
|
§ 2.2. The 42 conductors f coprime to three, (f,3) = 1
The members of a doublet do not necessarily have equal 3-class numbers 27.
This forces us to introduce three categories of doublets.
Category I: both members have 3-class group (3,9).
Category II: only a single member has 3-class number 27, the other has at least 81.
Category III: none of the members has 3-class group (3,9).
However, for Category I there arises an additional complication:
In general, the members with 3-class number 27 neither have the same TKT κ(K)
nor the same 3-class field tower group G32(K).
Therefore we use exponents denoting iteration.
|
No.
|
f
|
factors
|
TKT
|
κ(K)
|
G32(K)
|
|
1
|
4711
|
7*673
|
a.12
|
(0000)2
|
<81,3>2
|
|
2
|
5383
|
7*769
|
b.152
|
(0004)2
|
<243,14>2
|
|
3
|
11167
|
13*859
|
a.12
|
(0000)2
|
<81,3>2
|
|
4
|
12403
|
79*157
|
b.152
|
(0004)2
|
<243,14>2
|
|
5
|
12439
|
7*1777
|
a.12
|
(0000)2
|
<81,3>2
|
|
6
|
16177
|
7*2311
|
a.12
|
(0000)2
|
<81,3>2
|
|
7
|
17593
|
73*241
|
a.12
|
(0000)2
|
<81,3>2
|
|
9
|
21763
|
7*3109
|
b.15,A.20
|
(0004),(4444)
|
<243,13>,<729,12>
|
|
10
|
25963
|
7*3709
|
a.12
|
(0000)2
|
<81,3>2
|
|
11
|
27571
|
79*349
|
a.12
|
(0000)2
|
<81,3>2
|
|
12
|
28177
|
19*1483
|
b.152
|
(0004)2
|
<243,14>2
|
|
13
|
32311
|
79*409
|
b.152
|
(0004)2
|
<243,14>2
|
|
14
|
32689
|
97*337
|
a.12
|
(0000)2
|
<81,3>2
|
|
16
|
36667
|
37*991
|
E.12
|
(1234)
|
<729,17|20>↓
|
|
17
|
37933
|
7*5419
|
a.12
|
(0000)2
|
<81,3>2
|
|
18
|
38503
|
139*277
|
b.15,E.12
|
(0004),(1234)
|
<243,14>,<729,17|20>
|
|
19
|
40573
|
13*3121
|
a.12
|
(0000)2
|
<81,3>2
|
|
20
|
40873
|
7*5839
|
a.12
|
(0000)2
|
<81,3>2
|
|
21
|
41977
|
13*3229
|
a.1
|
(0000)
|
<243,15>
|
|
|
42127
|
103*409
|
|
|
|
|
22
|
42991
|
13*3307
|
a.1
|
(0000)
|
<243,15>
|
|
23
|
43081
|
67*643
|
a.12
|
(0000)2
|
<81,3>2
|
|
26
|
49849
|
79*631
|
b.15,E.12
|
(0004),(1234)
|
<243,14>,<729,17|20>
|
|
27
|
51847
|
139*373
|
a.12
|
(0000)2
|
<81,3>2
|
|
28
|
55657
|
7*7951
|
b.15,E.12
|
(0004),(1234)
|
<243,14>,<729,17|20>
|
|
29
|
55951
|
7*7993
|
a.12
|
(0000)2
|
<81,3>2
|
|
31
|
57811
|
13*4447
|
b.152
|
(0004)2
|
<243,13>2
|
|
33
|
59803
|
79*757
|
b.152
|
(0004)2
|
<243,14>2
|
|
34
|
59911
|
181*331
|
b.152
|
(0004)2
|
<243,13>2
|
|
35
|
62257
|
13*4789
|
a.12
|
(0000)2
|
<81,3>2
|
|
37
|
65383
|
151*433
|
a.12
|
(0000)2
|
<81,3>2
|
|
38
|
66829
|
7*9547
|
a.12
|
(0000)2
|
<81,3>2
|
|
39
|
68857
|
37*1861
|
b.15
|
(0004)
|
<243,14>↓
|
|
41
|
71611
|
19*3769
|
a.12
|
(0000)2
|
<81,3>2
|
|
43
|
73873
|
31*2383
|
a.12
|
(0000)2
|
<81,3>2
|
|
44
|
75859
|
7*10837
|
A.202
|
(4444)2
|
<729,12>2
|
|
45
|
77281
|
109*709
|
a.12
|
(0000)2
|
<81,3>2
|
|
48
|
84103
|
31*2713
|
b.152
|
(0004)2
|
<243,14>2
|
|
50
|
89863
|
73*1231
|
a.12
|
(0000)2
|
<81,3>2
|
|
51
|
94357
|
157*601
|
a.12
|
(0000)2
|
<81,3>2
|
|
53
|
96709
|
97*997
|
a.12
|
(0000)2
|
<81,3>2
|
|
54
|
96817
|
7*13831
|
a.12
|
(0000)2
|
<81,3>2
|
|
55
|
97249
|
79*1231
|
b.15
|
(0004)
|
<243,14>↓
|
Successful Conclusions
Our more than 10 years old theory of the Second 3-Genus Group
g = g32(K) = Gal(F31(K*)|Q)
is perfectly fitting together with the newly discovered structure of the
Second 3-Class Group
G = G32(K) = Gal(F32(K)|K)
of cyclic cubic base fields K whose conductor f is divisible by exactly
two primes (t = 2).
First we should point out that the abelianization g/g' ≅ Gal(K*|Q)
is of fixed type (3,3), since the 3-genus field K* of K is bicyclic bicubic,
whereas the abelianization G/G' ≅ Cl3(K) varies in dependence
on the cubic residue characters between the prime divisors of the conductor
and may be of type (3), (3,3), (3,9), (9,9), (9,27), …
having the structure of some nearly homocyclic abelian 3-group.
The possibilities for G provide a refinement of the classification by g
which is given in the following table:
|
cc(g)
|
g
|
g/g'
|
Parry matrix
|
cc(G)
|
G
|
G/G'
|
|
1
|
<9,2>
|
(3,3)
|
(1010) or (1121)
|
0
|
<3,1>
|
(3)
|
|
1
|
<27,3>
|
(3,3)
|
(1111) or (2121)
|
1
|
<9,2>
|
(3,3)
|
|
1
|
<81,7>
|
(3,3)
|
(1010)
|
1
|
<27,4>
|
(3,3)
|
|
2
|
<243,3>
|
(3,3)
|
(1001) or (1011)
|
2
|
<81,3>
|
(3,9)
|
|
2
|
<729,34|37>
|
(3,3)
|
(1010) or (1121) or (2121)
|
2
|
<243,14>
|
(3,9)
|
|
|
|
|
|
2
|
<243,13>
|
(3,9)
|
|
|
|
|
|
3
|
<729,12>
|
(3,9)
|
|
|
|
|
|
3
|
<729,17|20>
|
(3,9)
|
|
2
|
<729,40>
|
(3,3)
|
|
2
|
<243,15>
|
(3,9)
|
|
2
|
<729,40>↓
|
(3,3)
|
|
2
|
<243,14>↓
|
(3,9)
|
|
2
|
<2187,247>↓
|
(3,3)
|
|
3
|
<729,17|20>↓
|
(3,9)
|
|
3
|
<2187,64>↓
|
(3,3)
|
|
|
|
(9,9) or (9,27)
|
|
|
|
|
