Statistics and Minima:
-
Inspired by Amandine Leriche and Henri Cohen,
we use a very convenient parametrization of
generating polynomials for cyclic cubic fields
,
P(X) = X3 - 3*e*X - e*u,
and we apply our New Algorithm for determining the
Second p-Class Group
Gp2(K)
of a number field K
by computing the Kernels and Targets
of its Transfers to maximal subgroups.
We set p = 3 and obtain the complete
statistical evaluation of the smallest
cyclic cubic number fields K
whose conductors f are divisible by
two or three primes (2 ≤ t ≤ 3) and
whose 3-class groups are of type (3,3).
The distribution visualizes the population
of vertices on the coclass graphs G(3,1) and G(3,2)
of 3-groups of coclass either 1 or 2
by the groups G32(K).
-
For t = 3, we use graphs for visualizing
cubic residue characters between the
three prime divisors of the conductor f,
which were introduced by George Gras on p. 21 of
l-Classes of Ideals (Chapter VI, Section 3)
,
and the categories defined on p. 45 of Ayadi's Thesis
Cyclic Cubic Fields (Chapter 4, Section 4.2)
.
|
-
The Construction Process:
*************************************************************************
First Step: with the aid of
PARI/GP
we compute a list of
the first 250 conductors of cyclic cubic number fields K,
having a 3-class group Cl3(K) of type (3,3).
These conductors occur in the range 657 ≤ f ≤ 26467.
--------------------------------------------------------------
Second Step: by means of
MAGMA
we iterate through the list
of generating polynomials computed in the first step and determine
-
the 3-principalization kernel of K in its four
unramified cyclic cubic extensions N1,N2,N3,N4
(the transfer kernel type TKT),
-
the structure of the 3-class groups of N1,N2,N3,N4
(the transfer target type TTT).
--------------------------------------------------------------
Third Step: we use our unambiguous criteria
to identify the second 3-class group G32(K) of K
by the combination of TKT and TTT in the following table.
--------------------------------------------------------------
Remark for non-specialists: For 3-groups,
we use identifiers of the
SmallGroups Library
provided by
GAP
and
MAGMA
.
They are of the shape < order, counter >.
<81,7> is 3-Sylow subgroup of A9
and has an abelian maximal subgroup of type (3,3,3).
<9,2> ≅ (3,3) is the abelian root of the
unique coclass tree of coclass graph G(3,1)
consisting of 3-groups of maximal class.
The other groups are also vertices of G(3,1)
with three exceptions: <243,8>
in the stem of isoclinism family Φ6
which is root of one of the three coclass trees
with metabelian mainlines on coclass graph G(3,2)
consisting of 3-groups of second maximal class,
and the vertices <729,41> and <729,34…39>
at depth 1 of branch B(35) of
coclass tree T(<243,3>) on G(3,2).
G32(K)
|
TKT
|
Cl3(N1)
|
Cl3(N2)
|
Cl3(N3)
|
Cl3(N4)
|
Cl3(F31(K))
|
Coclass
|
<9,2>
|
(0,0,0,0)
|
(3)
|
(3)
|
(3)
|
(3)
|
1
|
1
|
<27,4>
|
(1,1,1,1)
|
(3,3)
|
(9)
|
(9)
|
(9)
|
(3)
|
1
|
<81,7>
|
(2,0,0,0)
|
(3,3,3)
|
(3,3)
|
(3,3)
|
(3,3)
|
(3,3)
|
1
|
<81,8>
|
(2,0,0,0)
|
(3,9)
|
(3,3)
|
(3,3)
|
(3,3)
|
(3,3)
|
1
|
<81,10>
|
(1,0,0,0)
|
(3,9)
|
(3,3)
|
(3,3)
|
(3,3)
|
(3,3)
|
1
|
<81,9>
|
(0,0,0,0)
|
(3,9)
|
(3,3)
|
(3,3)
|
(3,3)
|
(3,3)
|
1
|
<243,28…30>
|
(0,0,0,0)
|
(3,9)
|
(3,3)
|
(3,3)
|
(3,3)
|
(3,9)
|
1
|
<243,25>
|
(2,0,0,0)
|
(9,9)
|
(3,3)
|
(3,3)
|
(3,3)
|
(3,9)
|
1
|
<243,27>
|
(1,0,0,0)
|
(9,9)
|
(3,3)
|
(3,3)
|
(3,3)
|
(3,9)
|
1
|
<243,8>
|
(0,2,3,1)
|
(3,9)
|
(3,9)
|
(3,9)
|
(3,9)
|
(3,3,3)
|
2
|
<243,3>
|
(0,0,4,3)
|
(3,9)
|
(3,9)
|
(3,3,3)
|
(3,3,3)
|
(3,3,3)
|
2
|
<729,34…36>
|
(0,0,4,3)
|
(3,9)
|
(3,9)
|
(3,3,3)
|
(3,3,3)
|
(3,3,3,3)
|
2
|
<729,37…39>
|
(0,0,4,3)
|
(3,9)
|
(3,9)
|
(3,3,3)
|
(3,3,3)
|
(3,3,9)
|
2
|
<729,41>
|
(4,0,4,3)
|
(9,9)
|
(3,9)
|
(3,3,3)
|
(3,3,3)
|
(3,3,9)
|
2
|
Remarks:
1. Our new criteria use the TTT to locate a member
of an infinite sequence sharing a common TKT.
Of course, they would have been useless
in times before the availability of Pari/GP and Magma.
2. There are no selection rules in terms of branches of G(3,1)
for the groups G32(K) of cyclic cubic fields K.
(For real quadratic fields K of type (3,3)
we have proved that only every other branch is admissible.)
Therefore, cyclic cubic fields of type (3,3) reveal a
similar behavior as quadratic fields of type (2,2).
3. However, the Weak Leaf Conjecture seems to be satisfied
for cyclic cubic and quadratic fields of type (3,3).
The two instances which occurred up to now are
the distinction between <81,9> and its immediate descendants <243,28…30>
and between <243,3> and its immediate descendants <729,34…39>.
Descendants are of the same TKT but of higher defect of commutativity.
4. The decision between <81,9> and the indistinguishable batch
<243,28…30> closely approaches the limits
of current hardware and software technology.
This is one of the two instances where the 3-class group of
the Hilbert 3-class field F31(K) of K must be determined.
Since F31(K) is of absolute degree 27, the demand of memory is gigantic.
5. There seem to be differences in the behaviour of operating systems.
Whereas Windows 7 Pro strictly protects its own memory resources
and Magma frequently is forced to terminate complaining
"Cannot complete task, since OS doesn't grant access to required memory",
Mac OS X is obviously more generous and lets Magma use system memory.
Magma completes its task but the OS remains in an instable state
or even in a dead lock where Mac OS X doesn't show any reactions.
|
|
*************************************************************************
Statistics:
For the calculation of relative frequencies (percentages),
we need some absolute frequencies as references.
Due to the explanations given in the section "Details" below,
the 250 conductors in the range 657 ≤ f ≤ 26467 consist of
19 with t = 2 and 9 | f, corresponding to 38 fields,
69 with t = 2 and (f,3) = 1, corresponding to 138 fields,
since fields with t = 2 arise in doublets, and
84 with t = 3 and 9 | f, corresponding to 10*3 + 7*2 + 67*4 = 312 fields,
78 with t = 3 and (f,3) = 1, corresponding to 14*3 + 11*2 + 53*4 = 276 fields,
since fields with t = 3 may arise in triplets (Category I)
or in doublets (Category II) or in quartets (Category III).
G32(K)
|
<9,2>
|
<27,4>
|
<81,7>
|
<243,28…30>
|
<243,25>
|
<243,8>
|
<81,8>
|
<81,10>
|
<729,37…39>
|
<243,27>
|
<729,41>
|
<729,34…36>
|
t = 2, 9 | f
|
18/38 (47%)
|
20/38 (53%)
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
t = 2, (f,3) = 1
|
96/138 (70%)
|
42/138 (30%)
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
t = 3, 9 | f
|
220/312 (71%)
|
0
|
46/312 (15%)
|
11/312
|
4/312
|
6/312
|
6/312
|
12/312
|
0
|
3/312
|
2/312
|
2/312
|
t = 3, (f,3) = 1
|
184/276 (67%)
|
0
|
46/276 (17%)
|
2/276
|
1/276
|
15/276
|
8/276
|
16/276
|
4/276
|
0
|
0
|
0
|
The 3-tower consists of a single stage
if and only if G32(K) is abelian.
For t = 2, 9 | f,
the extraspecial group <27,4>
is populated most densely with 53%.
For t = 2, (f,3) = 1,
the single stage 3-towers
are dominating with 70%.
For t = 3, 9 | f,
the single stage 3-towers
are dominating with 71%.
For t = 3, (f,3) = 1,
the single stage 3-towers
are dominating with 67%.
Minimal Conductors:
G32(K)
|
<9,2>
|
<27,4>
|
<81,7>
|
<243,28…30>
|
<243,25>
|
<243,8>
|
<81,8>
|
<81,10>
|
<729,37…39>
|
<243,27>
|
<729,41>
|
<729,34…36>
|
t = 2, 9 | f
|
657
|
2439
|
|
|
|
|
|
|
|
|
|
|
t = 2, (f,3) = 1
|
1267
|
5971
|
|
|
|
|
|
|
|
|
|
|
t = 3, 9 | f
|
819
|
|
4599
|
4977
|
4977
|
21177
|
8001
|
8001
|
|
20367
|
22581
|
25929
|
t = 3, (f,3) = 1
|
1729
|
|
3913
|
10621
|
10621
|
7657
|
9709
|
9709
|
20293
|
|
|
|
Remarks:
1. The minimal conductors 657, 1267, 2439, 5971 were given
in the joint paper by
Ayadi, Azizi, Ismaïli, p. 474
.
2. In his
Thesis, p. 63 f.
,
Ayadi only indicated that the conductors
4977, 10621 with graph 1, and 7657, 8001 with graph 2
of category I give rise to triplets of 3-class rank 2
but he didn't determine the isomorphism type of G32(K).
|
|
*************************************************************************
Details for 250 conductors:
Although we split the conductors into 4 basic types,
the counting number No. refers to the
entire sequence, ordered by increasing values.
Cases with non-abelian group G32(K) are printed
in red boldface font for coclass 1,
in green boldface font for coclass 2.
§ 1. The 88 conductors f with two prime divisors, t = 2
According to the multiplicity formula m(f) = (3-1)t-1,
there are 2 cyclic cubic fields K sharing the common conductor f.
If one of them has 3-class group (3,3), then the same is true for the other.
Necessarily, the prime divisors of the conductor f
are mutual cubic residues with respect to each other.
Thus, we do not need graphs and categories in the case t = 2.
It turns out that both members of the doublet have the same TKT κ(K)
and the same 3-class field tower group G32(K),
which can be determined with the aid of the following theorem.
Theorem 1 (
Ayadi
,
1995).
Let f be a conductor divisible by exactly two primes, t = 2, such that
Cl3(K) ≅ (3,3) for both cyclic cubic fields K with conductor f.
Denote by n the number of prime divisors of the norm NK|Q(α) of
any non-trivial primitive ambiguous principal ideal (α) of any of the two fields K.
Then the second 3-class group G32(K) of both fields K
is given by <9,2> with TKT a.1, κ(K) = (0,0,0,0), if n = 2,
and by <27,4> with TKT A.1, κ(K) = (1,1,1,1), if n = 1.
Examples. We have
NK|Q(α) = 32*73, n = 2 for f = 657 = 32*73,
NK|Q(α) = 7*181, n = 2 for f = 1267 = 7*181,
NK|Q(α) = 271, n = 1 for f = 2439 = 32*271,
NK|Q(α) = 853, n = 1 for f = 5971 = 7*853,
according to
Ayadi, Azizi, Ismaïli, p. 474
.
§ 1.1. The 19 conductors f divisible by nine, 9 | f
No.
|
f
|
factors
|
TKT
|
κ(K)
|
G32(K)
|
1
|
657
|
32*73
|
a.1
|
(0000)
|
<9,2>
|
12
|
2439
|
32*271
|
A.1
|
(1111)
|
<27,4>
|
14
|
2763
|
32*307
|
a.1
|
(0000)
|
<9,2>
|
29
|
4707
|
32*523
|
A.1
|
(1111)
|
<27,4>
|
34
|
5193
|
32*577
|
a.1
|
(0000)
|
<9,2>
|
38
|
5517
|
32*613
|
a.1
|
(0000)
|
<9,2>
|
50
|
6813
|
32*757
|
a.1
|
(0000)
|
<9,2>
|
65
|
8271
|
32*919
|
A.1
|
(1111)
|
<27,4>
|
72
|
8919
|
32*991
|
A.1
|
(1111)
|
<27,4>
|
73
|
9081
|
32*1009
|
a.1
|
(0000)
|
<9,2>
|
84
|
10053
|
32*1117
|
A.1
|
(1111)
|
<27,4>
|
116
|
13779
|
32*1531
|
A.1
|
(1111)
|
<27,4>
|
118
|
13941
|
32*1549
|
A.1
|
(1111)
|
<27,4>
|
127
|
14589
|
32*1621
|
A.1
|
(1111)
|
<27,4>
|
141
|
16047
|
32*1783
|
a.1
|
(0000)
|
<9,2>
|
177
|
19611
|
32*2179
|
a.1
|
(0000)
|
<9,2>
|
182
|
20259
|
32*2251
|
A.1
|
(1111)
|
<27,4>
|
187
|
20583
|
32*2287
|
a.1
|
(0000)
|
<9,2>
|
194
|
21069
|
32*2341
|
A.1
|
(1111)
|
<27,4>
|
§ 1.2. The 69 conductors f coprime to three, (f,3) = 1
No.
|
f
|
factors
|
TKT
|
κ(K)
|
G32(K)
|
4
|
1267
|
7*181
|
a.1
|
(0000)
|
<9,2>
|
5
|
1339
|
13*103
|
a.1
|
(0000)
|
<9,2>
|
6
|
1561
|
7*223
|
a.1
|
(0000)
|
<9,2>
|
11
|
2359
|
7*337
|
a.1
|
(0000)
|
<9,2>
|
16
|
2869
|
19*151
|
a.1
|
(0000)
|
<9,2>
|
17
|
2947
|
7*421
|
a.1
|
(0000)
|
<9,2>
|
18
|
2977
|
13*229
|
a.1
|
(0000)
|
<9,2>
|
19
|
3241
|
7*463
|
a.1
|
(0000)
|
<9,2>
|
22
|
3811
|
37*103
|
a.1
|
(0000)
|
<9,2>
|
33
|
5053
|
31*163
|
a.1
|
(0000)
|
<9,2>
|
35
|
5263
|
19*277
|
a.1
|
(0000)
|
<9,2>
|
37
|
5473
|
13*421
|
a.1
|
(0000)
|
<9,2>
|
40
|
5677
|
7*811
|
a.1
|
(0000)
|
<9,2>
|
42
|
5971
|
7*853
|
A.1
|
(1111)
|
<27,4>
|
45
|
6181
|
7*883
|
a.1
|
(0000)
|
<9,2>
|
47
|
6487
|
13*499
|
a.1
|
(0000)
|
<9,2>
|
52
|
7087
|
19*373
|
A.1
|
(1111)
|
<27,4>
|
54
|
7147
|
7*1021
|
a.1
|
(0000)
|
<9,2>
|
57
|
7519
|
73*103
|
A.1
|
(1111)
|
<27,4>
|
59
|
7663
|
79*97
|
a.1
|
(0000)
|
<9,2>
|
63
|
8047
|
13*619
|
A.1
|
(1111)
|
<27,4>
|
66
|
8299
|
43*193
|
A.1
|
(1111)
|
<27,4>
|
67
|
8401
|
31*271
|
a.1
|
(0000)
|
<9,2>
|
76
|
9253
|
19*487
|
A.1
|
(1111)
|
<27,4>
|
83
|
9943
|
61*163
|
A.1
|
(1111)
|
<27,4>
|
90
|
10819
|
31*349
|
a.1
|
(0000)
|
<9,2>
|
91
|
10963
|
19*577
|
A.1
|
(1111)
|
<27,4>
|
92
|
11089
|
13*853
|
A.1
|
(1111)
|
<27,4>
|
98
|
11563
|
31*373
|
a.1
|
(0000)
|
<9,2>
|
101
|
12061
|
7*1723
|
a.1
|
(0000)
|
<9,2>
|
108
|
12931
|
67*193
|
a.1
|
(0000)
|
<9,2>
|
109
|
13117
|
13*1009
|
a.1
|
(0000)
|
<9,2>
|
110
|
13129
|
19*691
|
a.1
|
(0000)
|
<9,2>
|
117
|
13927
|
19*733
|
a.1
|
(0000)
|
<9,2>
|
120
|
14203
|
7*2029
|
A.1
|
(1111)
|
<27,4>
|
125
|
14521
|
13*1117
|
a.1
|
(0000)
|
<9,2>
|
128
|
14701
|
61*241
|
A.1
|
(1111)
|
<27,4>
|
134
|
15223
|
13*1171
|
a.1
|
(0000)
|
<9,2>
|
135
|
15577
|
37*421
|
a.1
|
(0000)
|
<9,2>
|
136
|
15613
|
13*1201
|
A.1
|
(1111)
|
<27,4>
|
137
|
15673
|
7*2239
|
A.1
|
(1111)
|
<27,4>
|
140
|
16021
|
37*433
|
a.1
|
(0000)
|
<9,2>
|
147
|
16897
|
61*277
|
A.1
|
(1111)
|
<27,4>
|
155
|
17587
|
43*409
|
a.1
|
(0000)
|
<9,2>
|
159
|
18019
|
37*487
|
a.1
|
(0000)
|
<9,2>
|
162
|
18319
|
7*2617
|
A.1
|
(1111)
|
<27,4>
|
169
|
18961
|
67*283
|
a.1
|
(0000)
|
<9,2>
|
171
|
19093
|
61*313
|
a.1
|
(0000)
|
<9,2>
|
172
|
19189
|
31*619
|
a.1
|
(0000)
|
<9,2>
|
174
|
19279
|
13*1483
|
a.1
|
(0000)
|
<9,2>
|
178
|
19651
|
43*457
|
a.1
|
(0000)
|
<9,2>
|
188
|
20671
|
7*2953
|
a.1
|
(0000)
|
<9,2>
|
197
|
21259
|
7*3037
|
a.1
|
(0000)
|
<9,2>
|
198
|
21451
|
19*1129
|
A.1
|
(1111)
|
<27,4>
|
199
|
21469
|
7*3067
|
a.1
|
(0000)
|
<9,2>
|
201
|
21691
|
109*199
|
a.1
|
(0000)
|
<9,2>
|
208
|
22141
|
7*3163
|
A.1
|
(1111)
|
<27,4>
|
211
|
22357
|
79*283
|
a.1
|
(0000)
|
<9,2>
|
213
|
22633
|
13*1741
|
a.1
|
(0000)
|
<9,2>
|
214
|
22849
|
73*313
|
a.1
|
(0000)
|
<9,2>
|
216
|
23233
|
7*3319
|
a.1
|
(0000)
|
<9,2>
|
217
|
23257
|
13*1789
|
A.1
|
(1111)
|
<27,4>
|
219
|
23383
|
67*349
|
A.1
|
(1111)
|
<27,4>
|
222
|
23611
|
7*3373
|
a.1
|
(0000)
|
<9,2>
|
230
|
24349
|
13*1873
|
A.1
|
(1111)
|
<27,4>
|
234
|
24703
|
7*3529
|
A.1
|
(1111)
|
<27,4>
|
236
|
25081
|
7*3583
|
a.1
|
(0000)
|
<9,2>
|
242
|
25699
|
31*829
|
a.1
|
(0000)
|
<9,2>
|
249
|
26359
|
43*613
|
a.1
|
(0000)
|
<9,2>
|
The conductors f ≤ 26359 in our Sections § 1.1 and § 1.2
overlap with those conductors f = p*q which are given in
Table 1 on p. 40 (p < 1000, q < 1000) and
Table 4 on p. 42 (p ≤ 151, q < 1000)
of the Section "Tables Numériques"
in the paper by
George Gras
.
We point out that Table 1 also contains cases
with class numbers divisible by 27
whereas Table 4 is restricted to 3-class groups of type (3,3).
|
|
§ 2. The 162 conductors f with three prime divisors, t = 3
According to the multiplicity formula m(f) = (3-1)t-1,
there are 4 cyclic cubic fields K sharing the common conductor f.
The members of a multiplet do not necessarily have equal 3-class ranks.
This forces us to introduce three categories of quadruplets.
Category I: three members have 3-class rank 2, the other has rank 3.
Category II: only two members have 3-class rank 2, the other two have rank 3.
Category III: all four members have 3-class rank 2.
However, there arises an additional complication:
In general, the members of rank 2 neither have the same TKT κ(K)
nor the same 3-class field tower group G32(K).
Therefore we use exponents denoting iteration.
A criterion for G32(K) to be abelian
is given in the following theorem [& conjecture].
Theorem 2 (
Ayadi
,
1995) [& Conjecture 2 (2013)].
Let f be a conductor divisible by exactly three primes, t = 3, such that
Cl3(K) ≅ (3,3) for all four cyclic cubic fields K with conductor f.
Then the second 3-class group G32(K) of all four fields K
is abelian of type <9,2> ≅ (3,3) with TKT a.1, κ(K) = (0,0,0,0), if
[and only if]
there are no mutual cubic residues among the prime divisors of f,
that is, f belongs to graphs 1,…,4 of category III.
[G32(K) is never abelian for categories I and II.]
For the group <81,7> with second largest density of population we suggest the following:
Conjecture 3 (2013).
Let f be a conductor divisible by exactly three primes, t = 3, such that
Cl3(K) ≅ (3,3) for only two of the four cyclic cubic fields K with conductor f,
that is, f belongs to graph 1 or 2 of category II.
Then the second 3-class group G32(K) of both fields K
is given by <81,7> ≅ Syl3(A9) with TKT a.3*, κ(K) = (2,0,0,0),
provided that f belongs to graph 2 when 9 | f.
§ 2.1. The 84 conductors f divisible by nine, 9 | f
No.
|
f
|
factors
|
category
|
graph
|
TKTs
|
κ(K)
|
G32(K)
|
2
|
819
|
32*7*13
|
III
|
2: 13 → 7
|
a.14
|
(0000)4
|
<9,2>4
|
3
|
1197
|
32*7*19
|
III
|
3: 7 → 19 → 9
|
a.14
|
(0000)4
|
<9,2>4
|
8
|
1953
|
32*7*31
|
III
|
1: δ ≠ 0
|
a.14
|
(0000)4
|
<9,2>4
|
9
|
2223
|
32*13*19
|
III
|
2: 19 → 9
|
a.14
|
(0000)4
|
<9,2>4
|
10
|
2331
|
32*7*37
|
III
|
2: 37 → 9
|
a.14
|
(0000)4
|
<9,2>4
|
13
|
2709
|
32*7*43
|
III
|
2: 43 → 7
|
a.14
|
(0000)4
|
<9,2>4
|
21
|
3627
|
32*13*31
|
III
|
2: 31 → 13
|
a.14
|
(0000)4
|
<9,2>4
|
23
|
3843
|
32*7*61
|
III
|
2: 9 → 61
|
a.14
|
(0000)4
|
<9,2>4
|
26
|
4221
|
32*7*67
|
III
|
2: 9 → 67
|
a.14
|
(0000)4
|
<9,2>4
|
27
|
4329
|
32*13*37
|
III
|
2: 37 → 9
|
a.14
|
(0000)4
|
<9,2>4
|
28
|
4599
|
32*7*73
|
III
|
7: 9 ↔ 73 ← 7
|
a.3*4
|
(2000)4
|
<81,7>4
|
31
|
4977
|
32*7*79
|
I
|
1: δ = 0
|
a.3,a.12
|
(2000),(0000)2
|
<243,25>,<243,28…30>2
|
32
|
5031
|
32*13*43
|
III
|
1: δ ≠ 0
|
a.14
|
(0000)4
|
<9,2>4
|
36
|
5301
|
32*19*31
|
III
|
3: 31 → 19 → 9
|
a.14
|
(0000)4
|
<9,2>4
|
44
|
6111
|
32*7*97
|
III
|
2: 97 → 7
|
a.14
|
(0000)4
|
<9,2>4
|
46
|
6327
|
32*19*37
|
II
|
2: 19 → 9 ← 37 → 19
|
a.3*2
|
(2000)2
|
<81,7>2
|
48
|
6489
|
32*7*103
|
III
|
2: 9 → 103
|
a.14
|
(0000)4
|
<9,2>4
|
51
|
6867
|
32*7*109
|
III
|
2: 109 → 9
|
a.14
|
(0000)4
|
<9,2>4
|
53
|
7137
|
32*13*61
|
III
|
2: 9 → 61
|
a.14
|
(0000)4
|
<9,2>4
|
56
|
7353
|
32*19*43
|
III
|
2: 19 → 9
|
a.14
|
(0000)4
|
<9,2>4
|
60
|
7839
|
32*13*67
|
III
|
2: 9 → 67
|
a.14
|
(0000)4
|
<9,2>4
|
61
|
8001
|
32*7*127
|
I
|
2: 9 ← 127 → 7
|
a.3,a.22
|
(2000),(1000)2
|
<81,8>,<81,10>2
|
68
|
8541
|
32*13*73
|
III
|
6: 9 ↔ 73 → 13
|
a.3*4
|
(2000)4
|
<81,7>4
|
69
|
8757
|
32*7*139
|
III
|
2: 139 → 7
|
a.14
|
(0000)4
|
<9,2>4
|
75
|
9243
|
32*13*79
|
III
|
2: 79 → 13
|
a.14
|
(0000)4
|
<9,2>4
|
79
|
9513
|
32*7*151
|
III
|
2: 9 → 151
|
a.14
|
(0000)4
|
<9,2>4
|
81
|
9891
|
32*7*157
|
III
|
2: 7 → 157
|
a.14
|
(0000)4
|
<9,2>4
|
85
|
10269
|
32*7*163
|
III
|
2: 163 → 9
|
a.14
|
(0000)4
|
<9,2>4
|
86
|
10323
|
32*31*37
|
III
|
3: 31 → 37 → 9
|
a.14
|
(0000)4
|
<9,2>4
|
87
|
10431
|
32*19*61
|
III
|
3: 19 → 9 → 61
|
a.14
|
(0000)4
|
<9,2>4
|
94
|
11349
|
32*13*97
|
I
|
1: δ = 0
|
a.3,a.22
|
(2000),(1000)2
|
<81,8>,<81,10>2
|
95
|
11403
|
32*7*181
|
III
|
6: 7 ↔ 181 → 9
|
a.3*4
|
(2000)4
|
<81,7>4
|
96
|
11457
|
32*19*67
|
III
|
3: 19 → 9 → 67
|
a.14
|
(0000)4
|
<9,2>4
|
99
|
11997
|
32*31*43
|
III
|
1: δ ≠ 0
|
a.14
|
(0000)4
|
<9,2>4
|
100
|
12051
|
32*13*103
|
III
|
7: 13 ↔ 103 ← 9
|
a.3*4
|
(2000)4
|
<81,7>4
|
102
|
12159
|
32*7*193
|
III
|
2: 9 → 193
|
a.14
|
(0000)4
|
<9,2>4
|
103
|
12483
|
32*19*73
|
III
|
7: 19 → 9 ↔ 73
|
a.3*4
|
(2000)4
|
<81,7>4
|
104
|
12537
|
32*7*199
|
III
|
2: 199 → 9
|
a.14
|
(0000)4
|
<9,2>4
|
106
|
12753
|
32*13*109
|
I
|
2: 9 ← 109 → 13
|
a.3,a.22
|
(2000),(1000)2
|
<81,8>,<81,10>2
|
112
|
13293
|
32*7*211
|
III
|
2: 211 → 7
|
a.14
|
(0000)4
|
<9,2>4
|
113
|
13509
|
32*19*79
|
III
|
2: 19 → 9
|
a.14
|
(0000)4
|
<9,2>4
|
119
|
14049
|
32*7*223
|
III
|
5: 7 ↔ 223
|
a.14
|
(0000)4
|
<243,28…30>4
|
122
|
14319
|
32*37*43
|
III
|
3: 43 → 37 → 9
|
a.14
|
(0000)4
|
<9,2>4
|
123
|
14427
|
32*7*229
|
III
|
1: δ ≠ 0
|
a.14
|
(0000)4
|
<9,2>4
|
130
|
14859
|
32*13*127
|
III
|
2: 127 → 9
|
a.14
|
(0000)4
|
<9,2>4
|
133
|
15183
|
32*7*241
|
III
|
1: δ ≠ 0
|
a.14
|
(0000)4
|
<9,2>4
|
142
|
16263
|
32*13*139
|
I
|
1: δ = 0
|
a.3,a.12
|
(2000),(0000)2
|
<243,25>,<243,28…30>2
|
145
|
16587
|
32*19*97
|
I
|
2: 9 ← 19 → 97
|
a.3,a.22
|
(2000),(1000)2
|
<81,8>,<81,10>2
|
148
|
17019
|
32*31*61
|
III
|
3: 9 → 61 → 31
|
a.14
|
(0000)4
|
<9,2>4
|
149
|
17073
|
32*7*271
|
III
|
5: 9 ↔ 271
|
a.3*4
|
(2000)4
|
<81,7>4
|
153
|
17451
|
32*7*277
|
III
|
1: δ ≠ 0
|
a.14
|
(0000)4
|
<9,2>4
|
156
|
17613
|
32*19*103
|
III
|
4: 19 → 9 → 103 → 19
|
a.14
|
(0000)4
|
<9,2>4
|
157
|
17667
|
32*13*151
|
III
|
3: 9 → 151 → 13
|
a.14
|
(0000)4
|
<9,2>4
|
158
|
17829
|
32*7*283
|
I
|
1: δ = 0
|
a.3,a.22
|
(2000),(1000)2
|
<81,8>,<81,10>2
|
164
|
18369
|
32*13*157
|
III
|
2: 157 → 13
|
a.14
|
(0000)4
|
<9,2>4
|
166
|
18639
|
32*19*109
|
II
|
2: 109 → 9 ← 19 → 109
|
a.3*2
|
(2000)2
|
<81,7>2
|
167
|
18693
|
32*31*67
|
III
|
2: 9 → 67
|
a.14
|
(0000)4
|
<9,2>4
|
170
|
19071
|
32*13*163
|
III
|
3: 13 → 163 → 9
|
a.14
|
(0000)4
|
<9,2>4
|
175
|
19341
|
32*7*307
|
III
|
6: 9 ↔ 307 → 7
|
a.3*4
|
(2000)4
|
<81,7>4
|
179
|
19719
|
32*7*313
|
III
|
2: 7 → 313
|
a.14
|
(0000)4
|
<9,2>4
|
184
|
20313
|
32*37*61
|
II
|
2: 61 → 9 ← 37 → 61
|
a.3*2
|
(2000)2
|
<81,7>2
|
185
|
20367
|
32*31*73
|
III
|
5: 9 ↔ 73
|
a.2,a.13
|
(1000),(0000)3
|
<243,27>,<243,28…30>3
|
191
|
20853
|
32*7*331
|
III
|
2: 7 → 331
|
a.14
|
(0000)4
|
<9,2>4
|
195
|
21177
|
32*13*181
|
I
|
2: 9 ← 181 → 13
|
c.213
|
(0231)3
|
<243,8>3
|
196
|
21231
|
32*7*337
|
III
|
5: 7 ↔ 337
|
a.22,a.32
|
(1000)2,(2000)2
|
<243,27>2,<243,25>2
|
202
|
21717
|
32*19*127
|
II
|
2: 9 → 127 ← 19 → 9
|
a.3*2
|
(2000)2
|
<81,7>2
|
206
|
21987
|
32*7*349
|
III
|
2: 349 → 7
|
a.14
|
(0000)4
|
<9,2>4
|
207
|
22041
|
32*19*97
|
I
|
1: δ = 0
|
a.3,a.22
|
(2000),(1000)2
|
<81,8>,<81,10>2
|
209
|
22311
|
32*37*67
|
III
|
3: 37 → 9 → 67
|
a.14
|
(0000)4
|
<9,2>4
|
212
|
22581
|
32*13*193
|
II
|
1: 9 → 193 ← 13
|
d.192
|
(4043)2
|
<729,41>2
|
215
|
23121
|
32*7*367
|
II
|
1: 7 → 367 ← 9
|
a.3*2
|
(2000)2
|
<81,7>2
|
218
|
23283
|
32*13*199
|
III
|
2: 199 → 9
|
a.14
|
(0000)4
|
<9,2>4
|
220
|
23499
|
32*7*373
|
III
|
2: 7 → 373
|
a.14
|
(0000)4
|
<9,2>4
|
221
|
23607
|
32*43*61
|
III
|
2: 9 → 61
|
a.14
|
(0000)4
|
<9,2>4
|
224
|
23769
|
32*19*139
|
III
|
2: 19 → 9
|
a.14
|
(0000)4
|
<9,2>4
|
226
|
23877
|
32*7*379
|
I
|
2: 7 ← 379 → 9
|
c.213
|
(0231)3
|
<243,8>3
|
229
|
24309
|
32*37*73
|
III
|
9: 37 → 9 ↔ 73 → 37
|
a.3*4
|
(2000)4
|
<81,7>4
|
233
|
24687
|
32*13*211
|
III
|
2: 13 → 211
|
a.14
|
(0000)4
|
<9,2>4
|
235
|
25011
|
32*7*397
|
III
|
2: 397 → 9
|
a.14
|
(0000)4
|
<9,2>4
|
244
|
25767
|
32*7*409
|
III
|
1: δ ≠ 0
|
a.14
|
(0000)4
|
<9,2>4
|
245
|
25821
|
32*19*151
|
III
|
9: 9 → 151 ↔ 19 → 9
|
a.3*4
|
(2000)4
|
<81,7>4
|
246
|
25929
|
32*43*67
|
II
|
1: 9 → 67 ← 43
|
b.102
|
(0043)2
|
<729,37…39>2
|
247
|
26091
|
32*13*223
|
III
|
2: 13 → 223
|
a.14
|
(0000)4
|
<9,2>4
|
248
|
26307
|
32*37*79
|
III
|
2: 37 → 9
|
a.14
|
(0000)4
|
<9,2>4
|
251
|
26523
|
32*7*421
|
III
|
5: 7 ↔ 421
|
a.2,a.13
|
(1000),(0000)3
|
<243,27>,<243,28…30>3
|
§ 2.2. The 78 conductors f coprime to three, (f,3) = 1
No.
|
f
|
factors
|
category
|
graph
|
TKTs
|
κ(K)
|
G32(K)
|
7
|
1729
|
7*13*19
|
III
|
3: 13 → 7 → 19
|
a.14
|
(0000)4
|
<9,2>4
|
15
|
2821
|
7*13*31
|
III
|
3: 31 → 13 → 7
|
a.14
|
(0000)4
|
<9,2>4
|
20
|
3367
|
7*13*37
|
III
|
2: 13 → 7
|
a.14
|
(0000)4
|
<9,2>4
|
24
|
3913
|
7*13*43
|
II
|
1: 13 → 7 ← 43
|
a.3*2
|
(2000)2
|
<81,7>2
|
25
|
4123
|
7*19*31
|
II
|
1: 7 → 19 ← 31
|
a.3*2
|
(2000)2
|
<81,7>2
|
30
|
4921
|
7*19*37
|
II
|
1: 7 → 19 ← 37
|
a.3*2
|
(2000)2
|
<81,7>2
|
39
|
5551
|
7*13*61
|
III
|
2: 13 → 7
|
a.14
|
(0000)4
|
<9,2>4
|
41
|
5719
|
7*19*43
|
III
|
3: 43 → 7 → 19
|
a.14
|
(0000)4
|
<9,2>4
|
43
|
6097
|
7*13*67
|
III
|
2: 13 → 7
|
a.14
|
(0000)4
|
<9,2>4
|
49
|
6643
|
7*13*73
|
III
|
4: 13 → 7 → 73 → 13
|
a.14
|
(0000)4
|
<9,2>4
|
55
|
7189
|
7*13*79
|
III
|
3: 79 → 13 → 7
|
a.14
|
(0000)4
|
<9,2>4
|
58
|
7657
|
13*19*31
|
I
|
2: 13 ← 31 → 19
|
c.213
|
(0231)3
|
<243,8>3
|
62
|
8029
|
7*31*37
|
III
|
2: 31 → 37
|
a.14
|
(0000)4
|
<9,2>4
|
64
|
8113
|
7*19*61
|
III
|
2: 7 → 19
|
a.14
|
(0000)4
|
<9,2>4
|
70
|
8827
|
7*13*97
|
II
|
1: 13 → 7 ← 97
|
a.3*2
|
(2000)2
|
<81,7>2
|
71
|
8911
|
7*19*67
|
III
|
2: 7 → 19
|
a.14
|
(0000)4
|
<9,2>4
|
74
|
9139
|
13*19*37
|
III
|
2: 37 → 19
|
a.14
|
(0000)4
|
<9,2>4
|
77
|
9331
|
7*31*43
|
III
|
2: 43 → 7
|
a.14
|
(0000)4
|
<9,2>4
|
78
|
9373
|
7*13*103
|
III
|
6: 7 ← 13 ↔ 103
|
a.3*4
|
(2000)4
|
<81,7>4
|
80
|
9709
|
7*19*73
|
I
|
2: 19 ← 7 → 73
|
a.3,a.22
|
(2000),(1000)2
|
<81,8>,<81,10>2
|
82
|
9919
|
7*13*109
|
III
|
3: 109 → 13 → 7
|
a.14
|
(0000)4
|
<9,2>4
|
88
|
10507
|
7*19*79
|
III
|
2: 7 → 19
|
a.14
|
(0000)4
|
<9,2>4
|
89
|
10621
|
13*19*43
|
I
|
1: δ = 0
|
a.3,a.12
|
(2000),(0000)2
|
<243,25>,<243,28…30>2
|
93
|
11137
|
7*37*43
|
I
|
2: 7 ← 43 → 37
|
a.3,a.22
|
(2000),(1000)2
|
<81,8>,<81,10>2
|
97
|
11557
|
7*13*127
|
II
|
1: 13 → 7 ← 127
|
a.3*2
|
(2000)2
|
<81,7>2
|
105
|
12649
|
7*13*139
|
II
|
1: 13 → 7 ← 139
|
a.3*2
|
(2000)2
|
<81,7>2
|
107
|
12901
|
7*19*97
|
III
|
4: 7 → 19 → 97 → 7
|
a.14
|
(0000)4
|
<9,2>4
|
111
|
13237
|
7*31*61
|
III
|
2: 61 → 31
|
a.14
|
(0000)4
|
<9,2>4
|
114
|
13699
|
7*19*103
|
II
|
1: 7 → 19 ← 103
|
a.3*2
|
(2000)2
|
<81,7>2
|
115
|
13741
|
7*13*151
|
III
|
3: 151 → 13 → 7
|
a.14
|
(0000)4
|
<9,2>4
|
121
|
14287
|
7*13*157
|
III
|
4: 7 → 157 → 13 → 7
|
a.14
|
(0000)4
|
<9,2>4
|
124
|
14497
|
7*19*109
|
III
|
3: 7 → 19 → 109
|
a.14
|
(0000)4
|
<9,2>4
|
126
|
14539
|
7*31*67
|
III
|
1: δ ≠ 0
|
a.14
|
(0000)4
|
<9,2>4
|
129
|
14833
|
7*13*163
|
I
|
2: 7 ← 13 → 163
|
a.3,a.22
|
(2000),(1000)2
|
<81,8>,<81,10>2
|
131
|
14911
|
13*31*37
|
I
|
2: 13 ← 31 → 37
|
a.3,a.22
|
(2000),(1000)2
|
<81,8>,<81,10>2
|
132
|
15067
|
13*19*61
|
III
|
1: δ ≠ 0
|
a.14
|
(0000)4
|
<9,2>4
|
138
|
15799
|
7*37*61
|
III
|
2: 37 → 61
|
a.14
|
(0000)4
|
<9,2>4
|
139
|
15841
|
7*31*73
|
III
|
2: 7 → 73
|
a.14
|
(0000)4
|
<9,2>4
|
143
|
16471
|
7*13*181
|
III
|
9: 13 → 7 ↔ 181 → 13
|
a.3*4
|
(2000)4
|
<81,7>4
|
144
|
16549
|
13*19*67
|
III
|
1: δ ≠ 0
|
a.14
|
(0000)4
|
<9,2>4
|
146
|
16891
|
7*19*127
|
III
|
4: 127 → 7 → 19 → 127
|
a.14
|
(0000)4
|
<9,2>4
|
150
|
17143
|
7*31*79
|
III
|
1: δ ≠ 0
|
a.14
|
(0000)4
|
<9,2>4
|
151
|
17329
|
13*31*43
|
III
|
2: 31 → 13
|
a.14
|
(0000)4
|
<9,2>4
|
152
|
17353
|
7*37*67
|
I
|
1: δ = 0
|
a.3,a.22
|
(2000),(1000)2
|
<81,8>,<81,10>2
|
154
|
17563
|
7*13*193
|
I
|
2: 7 ← 13 → 193
|
c.213
|
(0231)3
|
<243,8>3
|
160
|
18031
|
13*19*73
|
III
|
2: 73 → 13
|
a.14
|
(0000)4
|
<9,2>4
|
161
|
18109
|
7*13*199
|
III
|
2: 13 → 7
|
a.14
|
(0000)4
|
<9,2>4
|
163
|
18361
|
7*43*61
|
III
|
2: 43 → 7
|
a.14
|
(0000)4
|
<9,2>4
|
165
|
18487
|
7*19*139
|
III
|
3: 139 → 7 → 19
|
a.14
|
(0000)4
|
<9,2>4
|
168
|
18907
|
7*37*73
|
III
|
3: 7 → 73 → 37
|
a.14
|
(0000)4
|
<9,2>4
|
173
|
19201
|
7*13*211
|
II
|
2: 211 → 7 ← 13 → 211
|
a.3*2
|
(2000)2
|
<81,7>2
|
176
|
19513
|
13*19*79
|
III
|
2: 79 → 13
|
a.14
|
(0000)4
|
<9,2>4
|
180
|
20083
|
7*19*151
|
III
|
7: 7 → 19 ↔ 151
|
a.3*4
|
(2000)4
|
<81,7>4
|
181
|
20167
|
7*43*67
|
I
|
2: 7 ← 43 → 67
|
c.213
|
(0231)3
|
<243,8>3
|
183
|
20293
|
7*13*223
|
III
|
8: 13 → 7 ↔ 223 ← 13
|
b.104
|
(0043)4
|
<729,34…36>4
|
186
|
20461
|
7*37*79
|
III
|
1: δ ≠ 0
|
a.14
|
(0000)4
|
<9,2>4
|
189
|
20683
|
13*37*43
|
III
|
2: 43 → 37
|
a.14
|
(0000)4
|
<9,2>4
|
190
|
20839
|
7*13*229
|
III
|
6: 7 ← 13 ↔ 229
|
a.3*4
|
(2000)4
|
<81,7>4
|
192
|
20881
|
7*19*157
|
I
|
2: 19 ← 7 → 157
|
c.213
|
(0231)3
|
<243,8>3
|
193
|
21049
|
7*31*97
|
I
|
2: 7 ← 97 → 31
|
c.213
|
(0231)3
|
<243,8>3
|
200
|
21679
|
7*19*163
|
II
|
1: 7 → 19 ← 163
|
a.3*2
|
(2000)2
|
<81,7>2
|
203
|
21793
|
19*31*37
|
II
|
2: 37 → 19 ← 31 → 37
|
a.3*2
|
(2000)2
|
<81,7>2
|
204
|
21931
|
7*13*241
|
III
|
2: 13 → 7
|
a.14
|
(0000)4
|
<9,2>4
|
205
|
21973
|
7*43*73
|
II
|
2: 7 → 73 ← 43 → 7
|
a.3*2
|
(2000)2
|
<81,7>2
|
210
|
22351
|
7*31*103
|
III
|
2: 31 → 103
|
a.14
|
(0000)4
|
<9,2>4
|
223
|
23653
|
7*31*109
|
III
|
2: 109 → 31
|
a.14
|
(0000)4
|
<9,2>4
|
225
|
23779
|
7*43*79
|
III
|
2: 43 → 7
|
a.14
|
(0000)4
|
<9,2>4
|
227
|
23959
|
13*19*97
|
III
|
2: 19 → 97
|
a.14
|
(0000)4
|
<9,2>4
|
228
|
24073
|
7*19*181
|
III
|
9: 19 → 181 ↔ 7 → 19
|
a.3*4
|
(2000)4
|
<81,7>4
|
231
|
24583
|
13*31*61
|
III
|
3: 61 → 31 → 13
|
a.14
|
(0000)4
|
<9,2>4
|
232
|
24661
|
7*13*271
|
I
|
2: 7 ← 13 → 271
|
a.3,a.22
|
(2000),(1000)2
|
<81,8>,<81,10>2
|
237
|
25123
|
7*37*97
|
I
|
2: 7 ← 97 → 37
|
a.3,a.22
|
(2000),(1000)2
|
<81,8>,<81,10>2
|
238
|
25207
|
7*13*277
|
I
|
2: 7 ← 13 → 277
|
a.3,a.22
|
(2000),(1000)2
|
<81,8>,<81,10>2
|
239
|
25327
|
19*31*43
|
III
|
2: 31 → 19
|
a.14
|
(0000)4
|
<9,2>4
|
240
|
25441
|
13*19*103
|
III
|
6: 19 ← 103 ↔ 13
|
a.3*4
|
(2000)4
|
<81,7>4
|
241
|
25669
|
7*19*193
|
III
|
2: 7 → 19
|
a.14
|
(0000)4
|
<9,2>4
|
243
|
25753
|
7*13*283
|
III
|
2: 13 → 7
|
a.14
|
(0000)4
|
<9,2>4
|
250
|
26467
|
7*19*199
|
III
|
2: 7 → 19
|
a.14
|
(0000)4
|
<9,2>4
|
Remarks and Corrections:
1. The most exotic phenomena are the occurrences of conductors
f = 7657, 17563, 20167, 20881, 21049, resp. 20293, with (f,3) = 1,
and f = 21177, 23877, resp. 22581, resp. 25929, with 9 | f,
which give rise to second 3-class groups of coclass 2.
2. It is clear that the investigations have to be extended further
to see whether groups of coclass bigger than 2 are possible.
3. Due to theorems by
Blackburn
,
resp. by
Heider and Schmithals
,
the length of the 3-class field tower is exactly two,
for all cases having a non-abelian group G32(K),
except for <729,34…39> and <729,41>,
where the length is unknown and probably equal to three.
4. In the joint paper by
Ayadi, Azizi, Ismaïli, p. 474
,
there are two misprints and two errors
in the Numerical Examples after Theorem 4.4.
2 misprints:
Upper bound for the conductors is 6000 instead of 16000
and conductor 2843 should read 3843.
2 errors:
Conductor 3367 of category III, graph 2 is missing
from the listing of conductors in category III
having one of the graphs 1,2,3,4 and abelian G32(K).
Instead, the conductor 3913 of category II, graph 1
with non-abelian G32(K) is given erroneously.
This seems to be a systematic error, since 3913 doesn't
show up among the examples on p. 73 of
Ayadi's Thesis
either.
|
|
|
Explanation:
-
Since 2002, initiated by Aïssa Derhem,
we are planning this long desired extension
of
Mohammed Ayadi's Ph. D. Thesis
on 3-principalization in unramified cyclic extensions
of degree 3 over cyclic cubic fields K of type (3,3).
Except for the complete lack of statistical information,
the conductors divisible by two primes (t = 2) were settled
by a nice but rather simple theory which gives criteria
for only two possible second 3-class groups,
either G32(K) = <9,2> or <27,4>,
in terms of the ambiguous principal ideals of K.
See our presentation in
Joint Research 2002
.
However, here we show that the theory of
conductors divisible by three primes (t = 3)
is of considerable complexity, as expected,
and reveals a wealth of new configurations.
In spite of the complexity of possible cubic residue
conditions between the prime divisors of the conductor f,
nobody knew whether exotic variants of the
second 3-class groups G32(K) are to be expected.
Of course, we hoped that they will appear.
And, indeed, they did: all the groups G32(K)
different from <9,2> or <27,4>
were completely unknown up to now.
|