Historical introduction.
In his 1990 paper
[2]
,
Charles Parry used norms of primitive ambiguous principal ideals
in bicyclic bicubic fields k*
to calculate the index of all subfield units in the unit group Q* = (U:V)
of such an abelian field of 9th degree.
In his 1995 thesis
[3]
,
Mohammed Ayadi showed that capitulation phenomena
in
unramified cyclic cubic extensions of cyclic cubic fields
can be characterized by the norms of primitive ambiguous principal ideals
in the cyclic cubic fields.
|
|
*
|
Recent research results.
We have extended Ayadi's results to a broader class of cyclic cubic fields:
In the range 0 < f < 105 of conductors of cyclic cubic fields
there are 56 conductors divisible by 2 primes, f = p*q,
where p = 1(mod 3) or p = 32 and q = 1(mod 3)
(we put p' = 3 if p = 32 and p' = p otherwise),
such that the class numbers h-,h+ of the
corresponding 112 cyclic cubic fields
in couples (k-,k+) are divisible by 27.
They can be divided into 6 categories,
according to the automorphism group
G = Gal( (k*)1 | Q )
of the Hilbert 3-class field (k*)1
of the genus field k* = k- k+ over Q.
G is one of the 2-stage metabelian 3-groups,
which have been analyzed in Nebelung's thesis
[1]
.
They are called CBF-groups ("Cyclic or Bicyclic Factors"),
in view of their descending central series.
Each category can be characterized
by the structure of the 3-class group C* of k*,
by the Parry matrix
(1,0;0,1;x,y;w,z), shortly (x,y;w,z),
which describes the norms
b- = p'x qy, b+ = p'w qz
of primitive ambiguous principal ideals in k-,k+,
by the order of the capitulation kernel
cap = #ker(jk*|k-),
and by the index of the old units
Q* = (U:V)
in the class number relation
h* = Q* h- h+ hp hq / 35
resp.
h* = Q* h- h+ / 35
for the 3-contribution alone.
|
We are indebted to Aïssa Derhem for the theory
[4]
underlying the following classification
and we give a table
[5]
of details for all 56 conductors,
gratefully acknowledging that Karim Belabas has computed h* and C*
with the aid of PARI, assuming the truth of the Generalized Riemann Hypothesis (GRH).
| f
| p
| q
| (h-,h+)
| h*
| C*
| Q*
| G
| b-
| b+
| (x,y;w,z)
| (X,Y;W,Z)
|
| 4711 |
7 |
673 |
(27,27) |
27 |
[3, 3, 3] |
9 |
CBF 1a(4,5) |
72 |
6732 |
(2,0;0,2) |
(1,0;0,1) |
| 5383 |
7 |
769 |
(27,27) |
81 |
[9, 3, 3] |
27 |
CBF 1b(5,6) |
7692 |
7692 |
(0,2;0,2) |
(1,0;1,0) |
| 11167 |
13 |
859 |
(27,27) |
27 |
[3, 3, 3] |
9 |
CBF 1a(4,5) |
132 |
132*8592 |
(2,0;2,2) |
(1,0;1,1) |
| 12403 |
79 |
157 |
(27,27) |
81 |
[9, 3, 3] |
27 |
CBF 1b(5,6) |
792*157 |
792*1572 |
(2,1;2,2) |
(1,1;2,1) |
| 12439 |
7 |
1777 |
(27,27) |
432 |
[12, 12, 3] |
9 |
CBF 1a(4,5) |
7 |
7*1777 |
(1,0;1,1) |
(1,0;1,1) |
| 16177 |
7 |
2311 |
(27,27) |
108 |
[6, 6, 3] |
9 |
CBF 1a(4,5) |
72*2311 |
72*2311 |
(2,1;2,1) |
(2,1;2,1) |
| 17593 |
73 |
241 |
(189,27) |
189 |
[21, 3, 3] |
9 |
CBF 1a(4,5) |
241 |
73*241 |
(0,1;1,1) |
(1,0;1,1) |
| 20421 |
9 |
2269 |
(189,27) |
189 |
[21, 3, 3] |
9 |
CBF 1a(4,5) |
32 |
2269 |
(2,0;0,1) |
(1,0;0,1) |
| 21763 |
7 |
3109 |
(27,108) |
324 |
[6, 6, 3, 3] |
27 |
CBF 1b(5,6) |
7*3109 |
7*31092 |
(1,1;1,2) |
(1,1;2,1) |
| 25963 |
7 |
3709 |
(27,27) |
27 |
[3, 3, 3] |
9 |
CBF 1a(4,5) |
72 |
3709 |
(2,0;0,1) |
(1,0;0,1) |
| 27571 |
79 |
349 |
(27,27) |
108 |
[6, 6, 3] |
9 |
CBF 1a(4,5) |
792 |
3492 |
(2,0;0,2) |
(1,0;0,1) |
| 28177 |
19 |
1483 |
(27,108) |
324 |
[18, 6, 3] |
27 |
CBF 1b(5,6) |
192 |
|
(2,0; , ) |
|
| 32311 |
79 |
409 |
(27,108) |
324 |
[18, 6, 3] |
27 |
CBF 1b(5,6) |
4092 |
4092 |
(0,2;0,2) |
(1,0;1,0) |
| 32689 |
97 |
337 |
(27,108) |
108 |
[6, 6, 3] |
9 |
CBF 1a(4,5) |
97 |
97*337 |
(1,0;1,1) |
(1,0;1,1) |
| 35163 |
9 |
3907 |
(189,27) |
189 |
[21, 3, 3] |
9 |
CBF 1a(4,5) |
|
32*3907 |
( , ;2,1) |
|
| 36667 |
37 |
991 |
(27,243) |
729 |
[27, 9, 3] |
27 |
CBF 2b(7,8) |
372*991 |
372*9912 |
(2,1;2,2) |
(1,1;2,1) |
| 37933 |
7 |
5419 |
(27,27) |
27 |
[3, 3, 3] |
9 |
CBF 1a(4,5) |
5419 |
72*54192 |
(0,1;2,2) |
(1,0;1,1) |
| 38503 |
139 |
277 |
(27,108) |
1296 |
[18, 6, 6, 2] |
27 |
CBF 1b(5,6) |
139*2772 |
1392*2772 |
(1,2;2,2) |
(1,1;2,1) |
| 40573 |
13 |
3121 |
(27,189) |
189 |
[21, 3, 3] |
9 |
CBF 1a(4,5) |
13 |
3121 |
(1,0;0,1) |
(1,0;0,1) |
| 40873 |
7 |
5839 |
(27,27) |
27 |
[3, 3, 3] |
9 |
CBF 1a(4,5) |
58392 |
|
(0,2; , ) |
|
| 41977 |
13 |
3229 |
(81,189) |
567 |
[63, 3, 3] |
9 |
CBF 2a(5,6) |
32292 |
13*32292 |
(0,2;1,2) |
(1,0;2,1) |
| 42127 |
103 |
409 |
(243,81) |
729 |
[9, 9, 3, 3] |
9 |
CBF 2a(6,8) |
103*409 |
1032*4092 |
(1,1;2,2) |
(1,1;1,1) |
| 42991 |
13 |
3307 |
(27,81) |
81 |
[9, 3, 3] |
9 |
CBF 2a(5,6) |
132 |
3307 |
(2,0;0,1) |
(1,0;0,1) |
| 43081 |
67 |
643 |
(189,27) |
189 |
[21, 3, 3] |
9 |
CBF 1a(4,5) |
|
67*6432 |
( , ;1,2) |
|
| 44397 |
9 |
4933 |
(27,27) |
27 |
[3, 3, 3] |
9 |
CBF 1a(4,5) |
32 |
3*4933 |
(2,0;1,1) |
(1,0;1,1) |
| 49743 |
9 |
5527 |
(27,27) |
27 |
[3, 3, 3] |
9 |
CBF 1a(4,5) |
32 |
|
(2,0; , ) |
|
| 49849 |
79 |
631 |
(27,108) |
324 |
[18, 6, 3] |
27 |
CBF 1b(5,6) |
|
79*631 |
( , ;1,1) |
|
| 51847 |
139 |
373 |
(27,27) |
27 |
[3, 3, 3] |
9 |
CBF 1a(4,5) |
|
|
( , ; , ) |
|
| 55657 |
7 |
7951 |
(27,27) |
81 |
[9, 3, 3] |
27 |
CBF 1b(5,6) |
|
7*7951 |
( , ;1,1) |
|
| 55951 |
7 |
7993 |
(27,27) |
27 |
[3, 3, 3] |
9 |
CBF 1a(4,5) |
|
7*79932 |
( , ;1,2) |
|
| 56223 |
9 |
6247 |
(27,27) |
432 |
[12, 12, 3] |
9 |
CBF 1a(4,5) |
|
3*6247 |
( , ;1,1) |
|
| 57811 |
13 |
4447 |
(27,27) |
81 |
[3, 3, 3, 3] |
27 |
CBF 1b(5,6) |
44472 |
|
(0,2; , ) |
|
| 58329 |
9 |
6481 |
(27,189) |
189 |
[21, 3, 3] |
9 |
CBF 1a(4,5) |
|
32*6481 |
( , ;2,1) |
|
| 59803 |
79 |
757 |
(27,108) |
324 |
[18, 6, 3] |
27 |
CBF 1b(5,6) |
792 |
792 |
(2,0;2,0) |
(1,0;1,0) |
| 59911 |
181 |
331 |
(27,27) |
81 |
[3, 3, 3, 3] |
27 |
CBF 1b(5,6) |
|
|
( , ; , ) |
|
| 62257 |
13 |
4789 |
(108,351) |
5616 |
[78, 6, 6, 2] |
9 |
CBF 1a(4,5) |
47892 |
13*4789 |
(0,2;1,1) |
(1,0;1,1) |
| 64971 |
9 |
7219 |
(108,27) |
108 |
[6, 6, 3] |
9 |
CBF 1a(4,5) |
7219 |
|
(0,1; , ) |
|
| 65383 |
151 |
433 |
(27,108) |
108 |
[6, 6, 3] |
9 |
CBF 1a(4,5) |
|
|
( , ; , ) |
|
| 66829 |
7 |
9547 |
(108,189) |
756 |
[42, 6, 3] |
9 |
CBF 1a(4,5) |
|
7*9547 |
( , ;1,1) |
|
| 68857 |
37 |
1861 |
(81,108) |
972 |
[18, 18, 3] |
27 |
CBF 2b(6,7) |
37*1861 |
37*18612 |
(1,1;1,2) |
(1,1;2,1) |
| 69183 |
9 |
7687 |
(27,27) |
432 |
[6, 6, 6, 2] |
9 |
CBF 1a(4,5) |
|
3*7687 |
( , ;1,1) |
|
| 71611 |
19 |
3769 |
(27,108) |
108 |
[6, 6, 3] |
9 |
CBF 1a(4,5) |
|
19*3769 |
( , ;1,1) |
|
| 72099 |
9 |
8011 |
(189,27) |
756 |
[42, 6, 3] |
9 |
CBF 1a(4,5) |
32*8011 |
32*8011 |
(2,1;2,1) |
(2,1;2,1) |
| 73873 |
31 |
2383 |
(27,27) |
27 |
[3, 3, 3] |
9 |
CBF 1a(4,5) |
|
312*23832 |
( , ;2,2) |
|
| 75859 |
7 |
1083 |
(27,27) |
81 |
[3, 3, 3, 3] |
27 |
CBF 1b(5,6) |
|
|
( , ; , ) |
|
| 77281 |
109 |
709 |
(108,27) |
432 |
[6, 6, 6, 2] |
9 |
CBF 1a(4,5) |
|
109*709 |
( , ;1,1) |
|
| 78093 |
9 |
8677 |
(27,27) |
27 |
[3, 3, 3] |
9 |
CBF 1a(4,5) |
|
3*86772 |
( , ;1,2) |
|
| 81009 |
9 |
9001 |
(27,27) |
27 |
[3, 3, 3] |
9 |
CBF 1a(4,5) |
|
3*9001 |
( , ;1,1) |
|
| 84103 |
31 |
2713 |
(27,27) |
81 |
[9, 3, 3] |
27 |
CBF 1b(5,6) |
312 |
312 |
(2,0;2,0) |
(1,0;1,0) |
| 89109 |
9 |
9901 |
(189,27) |
189 |
[21, 3, 3] |
9 |
CBF 1a(4,5) |
|
32*9901 |
( , ;2,1) |
|
| 89863 |
73 |
1231 |
(27,27) |
27 |
[3, 3, 3] |
9 |
CBF 1a(4,5) |
|
732*12312 |
( , ;2,2) |
|
| 94357 |
157 |
601 |
(108,27) |
108 |
[6, 6, 3] |
9 |
CBF 1a(4,5) |
1572 |
|
(2,0; , ) |
|
| 95913 |
9 |
1065 |
(27,27) |
27 |
[3, 3, 3] |
9 |
CBF 1a(4,5) |
10657 |
3 |
(0,1;1,0) |
(1,0;0,1) |
| 96709 |
97 |
997 |
(27,27) |
27 |
[3, 3, 3] |
9 |
CBF 1a(4,5) |
997 |
972*9972 |
(0,1;2,2) |
(1,0;1,1) |
| 96817 |
7 |
1383 |
(27,27) |
108 |
[6, 6, 3] |
9 |
CBF 1a(4,5) |
|
|
( , ; , ) |
|
| 97249 |
79 |
1231 |
(81,27) |
243 |
[9, 9, 3] |
27 |
CBF 2b(6,7) |
|
79*1231 |
( , ;1,1) |
|
|
|
