# in cyclic cubic fields with conductor divisible by two primes.

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)

 References: [1] Brigitte Nebelung, Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3,3) und Anwendung auf das Kapitulationsproblem, Inauguraldissertation, Köln, 1989 [2] Charles J. Parry, Bicyclic Bicubic Fields, Canad. J. Math. 42 (1990), no. 3, 491 - 507 [3] Mohammed Ayadi, Sur la capitulation des 3-classes d'idéaux d'un corps cubique cyclique, Thèse de doctorat, Université Laval, Québec, 1995 [4] Aïssa Derhem, Sur les corps cubiques cycliques de conducteur divisible par deux premiers, Casablanca, 2002 [5] Daniel C. Mayer, Class Numbers and Principal Factorizations of Families of Cyclic Cubic Fields with Discriminant d < 1010, (Latest Update) Univ. Graz, Computer Centre, 2008

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