Joint research 2002

of Karim Belabas, Aïssa Derhem, and Daniel C. Mayer

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On these pages, we present most recent results of our joint research, directly from the lab.
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Basic bibliography:
K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 1213-1237
A. Derhem, Capitulation dans les extensions quadratiques non ramifiées de corps de nombres cubiques cycliques,
Thèse de doctorat, Université Laval, Québec, 1988
D. C. Mayer, Multiplicities of dihedral discriminants, Math. Comp. 58 (1992), 831-847 and S55-S58
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Web master's e-mail address:
contact@algebra.at
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Ambiguous Principal Ideals in Cyclic Cubic Fields (2002/09/18)
Dan (02/09/17):
Ambiguous principal ideals have proved to be a valuable tool
for investigating a relative extension L | k of algebraic number fields:

1. The order of the group of non-trivial ambiguous principal ideals
enters estimates of the class number,
hL = (IL:PL) is divisible by (and thus >=) hk * #(ILG / Ik) / #(PLG / Pk).

2. The group of non-trivial ambiguous principal ideals is isomorphic
to the 1st cohomology group of G with unit coefficients,
PLG / Pk = H1(G,UL) (Iwasawa), where G = Gal(L|k).

3. Generators of ambiguous principal ideals can be used
to construct units and to compute regulators,
e. g., e = a3/Norm(a), R = log(e) = 3log(a) - log(Norm(a)),
for a complex cubic field L | Q.
New applications of ambiguous principal ideals have been found
by Charles Parry and Mohammed Ayadi:

4. Parry used norms of primitive ambiguous principal ideals
in his 1990 paper [2] on bicyclic bicubic fields
to calculate the index of all subfield units in the unit group
of such an abelian 9th degree field.

5. In his 1995 thesis [3] , Ayadi showed that capitulation phenomena in
unramified cubic extensions of cyclic cubic fields
can be characterized by the norms of primitive ambiguous principal ideals
in the cyclic cubic fields.
Warning:
It should be pointed out that different designations have developed
for Norms of Primitive Ambiguous Principal Ideals over the time:

a) Barrucand and Cohn called them Principal Factors of the Discriminant in 1969,
b) I used the concept of Differential Principal Factors in 1988,
c) Derhem and Couture invented the name Kummer Resolvents in 1988,
d) Ayadi called them Parry Constants or Parry Invariants in 1995.
Dan (02/09/17):
Now we extend Ayadi's results to a broader class of cyclic cubic fields:

In the range 0 < f < 10^5 of conductors
there are 56 of them divisible by 2 primes, f = p*q,
where p = 1(mod 3) or p = 32 and q = 1(mod 3)
and where 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 metabelian groups,
which have been analyzed in Nebelung's thesis [1] .
They are called ZEF-groups ("Rang Zwei oder Eins Faktoren"),
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 q^y, b+ = p'^w q^z
of primitive ambiguous principal ideals in k-,k+,
by the order of the capitulation kernel
cap = #ker(j_(k*|k-)),
and by the index of the old units
Q* = (U:V)
in the class number relation
h* = Q* h- h+ h_p h_q / 3^5
resp.
h* = Q* h- h+ / 3^5
for the 3-contribution alone.
Dan (02/09/17):
I am indebted to Aïssa Derhem for the theory [4]
underlying the following classification.
In the structure symbol for the descending central series of G,
"=" means a factor of 3-rank 2, and "-" means a factor of 3-rank 1.
Whereas G was of maximal class (e = 2) in Ayadi's results, we now have G of

Second maximal class (e = 3)

1st category: group of class 3,
G = G^(4,5)((0,0,0,0),0) in ZEF Ia(4,5)
structure: G_1 = G_2 - G_3 = G_4
C* = (3,3,3)
(x,y;w,z) = (1,0;0,1) or (1,0;1,1) or (2,1;2,1)
cap = 9
Q* = 9
[ occurs for 74 fields ]

2nd category: group of class 4,
G = G^(5,6)((0,0,0,0),+-1) in ZEF Ib(5,6)
structure: G_1 = G_2 - G_3 = G_4 - G_5
C* = (9,3,3) or (3,3,3,3)
(x,y;w,z) = (1,0;1,0) or (1,1;2,1)
[ but no correlation between C* and (x,y;w,z) ]
cap = 3
Q* = 27
[ occurs for 26 fields ]

3rd category: group of class 4,
G = G^(5,6)((0,0,0,0),0) in ZEF IIa(5,6)
structure: G_1 = G_2 - G_3 = G_4 - G_5
C* = (9,3,3)
(x,y;w,z) = (1,0;0,1) or (1,0;2,1)
cap = 9
Q* = 9
[ occurs for 4 fields with f = 41977,42991 ]

4th category: group of class 5,
G = G^(6,7)((0,0,0,0),+-1) in ZEF IIb(6,7)
structure: G_1 = G_2 - G_3 = G_4 - G_5 - G_6
C* = (9,9,3)
(x,y;w,z) = (1,1;2,1)
cap = 3
Q* = 27
[ occurs for 4 fields with f = 68857,97249 ]

5th category: group of class 6,
G = G^(7,8)((0,0,0,0),+-1) in ZEF IIb(7,8)
structure: G_1 = G_2 - G_3 = G_4 - G_5 - G_6 - G_7
C* = (27,9,3)
(x,y;w,z) = (1,1;2,1)
cap = 3
Q* = 27
[ occurs for 2 fields with f = 36667 ]
Or even G of

Lower than second maximal class (e = 4)

6th category: group of class 5,
G = G^(6,8)((0,0,0,0),0) in ZEF IIa(6,8)
structure: G_1 = G_2 - G_3 = G_4 = G_5 - G_6
C* = (9,9,3,3)
(x,y;w,z) = (1,1;1,1)
cap = 9
Q* = 9
[ occurs for 2 fields with f = 42127 ]
Dan (02/09/18):
Finally I 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 ZEF 1a(4,5) 7^2 673^2 (2,0;0,2) (1,0;0,1)
5383 7 769 (27,27) 81 [9, 3, 3] 27 ZEF 1b(5,6) 769^2 769^2 (0,2;0,2) (1,0;1,0)
11167 13 859 (27,27) 27 [3, 3, 3] 9 ZEF 1a(4,5) 13^2 13^2*859^2 (2,0;2,2) (1,0;1,1)
12403 79 157 (27,27) 81 [9, 3, 3] 27 ZEF 1b(5,6) 79^2*157 79^2*157^2 (2,1;2,2) (1,1;2,1)
12439 7 1777 (27,27) 432 [12, 12, 3] 9 ZEF 1a(4,5) 7 7*1777 (1,0;1,1) (1,0;1,1)
16177 7 2311 (27,27) 108 [6, 6, 3] 9 ZEF 1a(4,5) 7^2*2311 ( , ;2,1)
17593 73 241 (189,27) 189 [21, 3, 3] 9 ZEF 1a(4,5) 241 73*241 (0,1;1,1) (1,0;1,1)
20421 9 2269 (189,27) 189 [21, 3, 3] 9 ZEF 1a(4,5) 3^2 2269 (2,0;0,1) (1,0;0,1)
21763 7 3109 (27,108) 324 [6, 6, 3, 3] 27 ZEF 1b(5,6) 7*3109 7*3109^2 (1,1;1,2) (1,1;2,1)
25963 7 3709 (27,27) 27 [3, 3, 3] 9 ZEF 1a(4,5) 7^2 3709 (2,0;0,1) (1,0;0,1)
27571 79 349 (27,27) 108 [6, 6, 3] 9 ZEF 1a(4,5) 79^2 (2,0; , )
28177 19 1483 (27,108) 324 [18, 6, 3] 27 ZEF 1b(5,6) 19^2 (2,0; , )
32311 79 409 (27,108) 324 [18, 6, 3] 27 ZEF 1b(5,6) 409^2 409^2 (0,2;0,2) (1,0;1,0)
32689 97 337 (27,108) 108 [6, 6, 3] 9 ZEF 1a(4,5) 97 97*337 (1,0;1,1) (1,0;1,1)
35163 9 3907 (189,27) 189 [21, 3, 3] 9 ZEF 1a(4,5) 3^2*3907 ( , ;2,1)
36667 37 991 (27,243) 729 [27, 9, 3] 27 ZEF 2b(7,8) 37^2*991 37^2*991^2 (2,1;2,2) (1,1;2,1)
37933 7 5419 (27,27) 27 [3, 3, 3] 9 ZEF 1a(4,5) 5419 7^2*5419^2 (0,1;2,2) (1,0;1,1)
38503 139 277 (27,108) 1296 [18, 6, 6, 2] 27 ZEF 1b(5,6) 139*277^2 139^2*277^2 (1,2;2,2) (1,1;2,1)
40573 13 3121 (27,189) 189 [21, 3, 3] 9 ZEF 1a(4,5) 13 3121 (1,0;0,1) (1,0;0,1)
40873 7 5839 (27,27) 27 [3, 3, 3] 9 ZEF 1a(4,5) 5839^2 (0,2; , )
41977 13 3229 (81,189) 567 [63, 3, 3] 9 ZEF 2a(5,6) 3229^2 13*3229^2 (0,2;1,2) (1,0;2,1)
42127 103 409 (243,81) 729 [9, 9, 3, 3] 9 ZEF 2a(6,8) 103*409 103^2*409^2 (1,1;2,2) (1,1;1,1)
42991 13 3307 (27,81) 81 [9, 3, 3] 9 ZEF 2a(5,6) 13^2 3307 (2,0;0,1) (1,0;0,1)
43081 67 643 (189,27) 189 [21, 3, 3] 9 ZEF 1a(4,5) 67*643^2 ( , ;1,2)
44397 9 4933 (27,27) 27 [3, 3, 3] 9 ZEF 1a(4,5) 3^2 3*4933 (2,0;1,1) (1,0;1,1)
49743 9 5527 (27,27) 27 [3, 3, 3] 9 ZEF 1a(4,5) 3^2 (2,0; , )
49849 79 631 (27,108) 324 [18, 6, 3] 27 ZEF 1b(5,6) 79*631 ( , ;1,1)
51847 139 373 (27,27) 27 [3, 3, 3] 9 ZEF 1a(4,5) ( , ; , )
55657 7 7951 (27,27) 81 [9, 3, 3] 27 ZEF 1b(5,6) 7*7951 ( , ;1,1)
55951 7 7993 (27,27) 27 [3, 3, 3] 9 ZEF 1a(4,5) 7*7993^2 ( , ;1,2)
56223 9 6247 (27,27) 432 [12, 12, 3] 9 ZEF 1a(4,5) 3*6247 ( , ;1,1)
57811 13 4447 (27,27) 81 [3, 3, 3, 3] 27 ZEF 1b(5,6) 4447^2 (0,2; , )
58329 9 6481 (27,189) 189 [21, 3, 3] 9 ZEF 1a(4,5) 3^2*6481 ( , ;2,1)
59803 79 757 (27,108) 324 [18, 6, 3] 27 ZEF 1b(5,6) 79^2 79^2 (2,0;2,0) (1,0;1,0)
59911 181 331 (27,27) 81 [3, 3, 3, 3] 27 ZEF 1b(5,6) ( , ; , )
62257 13 4789 (108,351) 5616 [78, 6, 6, 2] 9 ZEF 1a(4,5) 4789^2 13*4789 (0,2;1,1) (1,0;1,1)
64971 9 7219 (108,27) 108 [6, 6, 3] 9 ZEF 1a(4,5) 7219 (0,1; , )
65383 151 433 (27,108) 108 [6, 6, 3] 9 ZEF 1a(4,5) ( , ; , )
66829 7 9547 (108,189) 756 [42, 6, 3] 9 ZEF 1a(4,5) 7*9547 ( , ;1,1)
68857 37 1861 (81,108) 972 [18, 18, 3] 27 ZEF 2b(6,7) 37*1861 37*1861^2 (1,1;1,2) (1,1;2,1)
69183 9 7687 (27,27) 432 [6, 6, 6, 2] 9 ZEF 1a(4,5) 3*7687 ( , ;1,1)
71611 19 3769 (27,108) 108 [6, 6, 3] 9 ZEF 1a(4,5) 19*3769 ( , ;1,1)
72099 9 8011 (189,27) 756 [42, 6, 3] 9 ZEF 1a(4,5) 3^2*8011 3^2*8011 (2,1;2,1) (2,1;2,1)
73873 31 2383 (27,27) 27 [3, 3, 3] 9 ZEF 1a(4,5) 31^2*2383^2 ( , ;2,2)
75859 7 1083 (27,27) 81 [3, 3, 3, 3] 27 ZEF 1b(5,6) ( , ; , )
77281 109 709 (108,27) 432 [6, 6, 6, 2] 9 ZEF 1a(4,5) 109*709 ( , ;1,1)
78093 9 8677 (27,27) 27 [3, 3, 3] 9 ZEF 1a(4,5) 3*8677^2 ( , ;1,2)
81009 9 9001 (27,27) 27 [3, 3, 3] 9 ZEF 1a(4,5) 3*9001 ( , ;1,1)
84103 31 2713 (27,27) 81 [9, 3, 3] 27 ZEF 1b(5,6) 31^2 31^2 (2,0;2,0) (1,0;1,0)
89109 9 9901 (189,27) 189 [21, 3, 3] 9 ZEF 1a(4,5) 3^2*9901 ( , ;2,1)
89863 73 1231 (27,27) 27 [3, 3, 3] 9 ZEF 1a(4,5) 73^2*1231^2 ( , ;2,2)
94357 157 601 (108,27) 108 [6, 6, 3] 9 ZEF 1a(4,5) 157^2 (2,0; , )
95913 9 1065 (27,27) 27 [3, 3, 3] 9 ZEF 1a(4,5) 10657 3 (0,1;1,0) (1,0;0,1)
96709 97 997 (27,27) 27 [3, 3, 3] 9 ZEF 1a(4,5) 997 97^2*997^2 (0,1;2,2) (1,0;1,1)
96817 7 1383 (27,27) 108 [6, 6, 3] 9 ZEF 1a(4,5) ( , ; , )
97249 79 1231 (81,27) 243 [9, 9, 3] 27 ZEF 2b(6,7) 79*1231 ( , ;1,1)
It is adequate to give some remarks concerning the Parry Matrix,
(1,0;0,1;x,y;w,z).

1. I always omit the trivial part concerning k_p and k_q,
since the norms of primitive ambiguous principal ideals in
these fields are uniquely determined:
k_p has { p', p'^2 } and k_q has { q, q^2 }.
So the Parry Invariant (minimal norm) is
p' for k_p and q for k_q.
Thus the trivial part is (1,0;0,1), which also represents
(1,0;0,2),(2,0;0,1), and (2,0;0,2).

2. For k- and k+, the norms of primitive ambiguous principal ideals
form a unique set among the 4 possibilities
{ p', p'^2 }, { q, q^2 }, { p'q, p'^2q^2 }, { p'^2q, p'q^2 }.
With the normalization p' < q, the corresponding Parry Invariants
are p', q, p'q, p'^2q.

Canonical Representatives (X,Y;W,Z) for the essential part (x,y;w,z) are therefore:

a) (1,0;1,0) (equal single primes) for
(1,0;2,0), (2,0;1,0), and (2,0;2,0)
and for symmetry reasons also for
(0,1;0,1), (0,1;0,2), (0,2;0,1), (0,2;0,2)

b) (1,0;0,1) (different single primes) for
(1,0;0,2), (2,0;0,1), and (2,0;0,2)
and for symmetry reasons also for
(0,1;1,0), (0,1;2,0), (0,2;1,0), (0,2;2,0)

c) (1,0;1,1) (inhomogeneous mixed case) for
(1,0;2,2), (2,0;1,1), and (2,0;2,2)
and for symmetry reasons also for
(0,1;1,1), (0,1;2,2), (0,2;1,1), (0,2;2,2)
(1,1;1,0), (1,1;2,0), (2,2;1,0), (2,2;2,0)
(1,1;0,1), (1,1;0,2), (2,2;0,1), (2,2;0,2)

d) (1,0;2,1) (homogeneous mixed case) for
(1,0;1,2), (2,0;2,1), and (2,0;1,2)
and for symmetry reasons also for
(0,1;2,1), (0,1;1,2), (0,2;2,1), (0,2;1,2)
(2,1;1,0), (2,1;2,0), (1,2;1,0), (1,2;2,0)
(2,1;0,1), (2,1;0,2), (1,2;0,1), (1,2;0,2)

e) (1,1;1,1) (inhomogeneous equal double primes) for
(1,1;2,2), (2,2;1,1), and (2,2;2,2)

f) (2,1;2,1) (homogeneous equal double primes) for
(2,1;1,2), (1,2;2,1), and (1,2;1,2)

g) (1,1;2,1) (different double primes) for
(1,1;1,2), (2,2;2,1), and (2,2;1,2)
and for symmetry reasons also for
(2,1;1,1), (2,1;2,2), (1,2;1,1), (1,2;2,2)

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,
Univ. Graz, Computer Centre, 2002

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