Memorial 2009



Dedicated to the Memory of

Alexander Aigner

in the Year 2009:

All known Examples for Principalization Types.

1. Definition of the Principalization Type.

Assume that p ≥ 2 denotes an arbitrary rational prime.

Let K be a base field with p-class rank 2 and
with p-class group Sylp(C(K)) of p-elementary abelian type (p,p).
Denote by K1 the 1st Hilbert p-class field of K.
According to the Artin Reciprocity Law of Class Field Theory,
the p-class group Sylp(C(K)) ≅ (p,p) of K is isomorphic to the
relative automorphism group Gal(K1|K) of K1 over K.
In particular, the p+1 cyclic subgroups Ci (1 ≤ i ≤ p+1) of Sylp(C(K)) are mapped to the
Galois groups Gal(K1|Ni) = Mi of p+1 unramified cyclic extensions Ni of K of p-th degree.
In fact, the Ci = Norm(Ni|K)Sylp(C(Ni)) are relative norms of p-class groups.
The following diagrams illustrate the Galois correspondence and the Artin isomorphism:

K1 = N0
//\\
N1N2...Np+1
\\//
K
Gal(K1|K1) = 1
/
|
\
M1
Gal(K1|Ni) = Mi
Mp+1
\
|
/
Gal(K1|K)
1
//\\
C1C2...Cp+1
\\//
Sylp(C(K)) = C0

Now we consider the class extension homomorphisms j(Ni|K): Sylp(C(K)) → Sylp(C(Ni)).
We say an ideal class of K that is mapped to the principal class 1 of Ni by j(Ni|K)
principalizes or becomes principal or capitulates in Ni.

According to Hilbert's Theorem 94, none of the principalization kernels Kern j(Ni|K) is trivial,
provided that Ni|K is an extension with odd prime degree p ≥ 3,
since there is an isomorphism to the unit cohomology of N = Ni:
Kern j(N|K) = PN<S>/PK ≅ EN|K/UNS-1 > 1,
where Gal(N|K) = <S> and EN|K denotes the intersection of UN with Kern Norm(N|K).

According to the Hilbert/Artin/Furtwängler Principal Ideal Theorem,
we have complete principalization in the Hilbert p-class field K1 = N0:
Kern j(N0|K) = Sylp(C(K)) = C0.

Thus, there are p+2 possibilities for each Kern j(Ni|K), C0,C1,…,Cp+1.

If p ≥ 3 is an odd rational prime and K is a quadratic field,
then we define a Natural Principalization Type (k(1),…,k(p+1)),
ordering the Ni (1 ≤ i ≤ p+1), which have dihedral absolute groups Gal(Ni|Q) = D(2p),
by increasing regulators of their absolute subfields Li of p-th degree:
for each index 0 ≤ i ≤ p+1 there exists a unique index 0 ≤ k(i) ≤ p+1
such that Kern j(Ni|K) = Ck(i) (in particular, we always have k(0) = 0).

2. All known Examples for Principalization Types.

Only very few results [2, Beispiel 5, p. 22, f.] are known
about the p-principalization for odd primes p ≥ 5.

Consequently we restrict ourselves to the case p = 3 and we give all results about
the 3-principalization in unramified cyclic cubic extensions of quadratic base fields
with 3-class rank 2 and 3-class group of type (3,3).

2.1. 3-Principalization over Complex Quadratic Fields.

In the following tables, we give the natural principalization type (k(1)…k(4))
of complex quadratic fields K = Q(d1/2) with discriminant -700000 < d < 0
and 3-class group of type (3,3).

The range -50000 < d < 0 is covered completely.
Our computations of 2003 for 42 new cases in the range -50000 < d < -30000 [7]
extend our own results for 22 fields with -30000 < d < -20000 of 1989 [5]
and the 13 examples with -20000 < d < 0 by Heider and Schmithals in 1982 [2] .

The results by Brink in his 1984 thesis [3] came to our knowledge with delay in 2006,
since they are not mentioned explicitly in the paper [4] by Brink and Gold.

As a supplement, we give some particularly interesting singular cases
with associated 2-stage metabelian 3-groups [9] of exceptionally high order
in the range -700000 < d < -50000 [8,10,11] .

Rare or exceptional cases are printed in boldface digits.

No.dK(k(1)…k(4))TypeRef.
1-3896(4111)H.4[2]
2-4027(2331)D.10[1]
3-6583(4111)H.4[2]
4-8751(1421)D.10[2]
5-9748(2214)E.9[1]
---------------------------
6-12067(4321)G.19[2]
7-12131(2244)D.5[2]
8-15544(1122)E.6[2]
9-16627(2313)E.14[2]
10-17131(3214)G.16[2]
11-18555(1313)E.6[2]
12-19187(4334)D.5[2]
13-19651(1312)D.10[2]
---------------------------
14-20276(3232)D.5[3,5]
15-20568(1414)D.5[3,5]
16-21224(1421)D.10[3,5]
17-21668(2111)H.4.V1[3,5]
18-22395(1132)E.9[3,5]
19-22443(2214)E.9[3,5]
20-22711(2331)D.10[3,5]
21-23428(3323)H.4[3,5]
22-23683(3223)E.6[3,5]
23-24340(1133)D.5[3,5]
24-24884(2134)G.16[3,5]
25-24904(2241)D.10[3,5]
26-25447(3343)H.4[5]
27-26139(3221)D.10[5]
28-26760(1133)D.5[3,5]
29-27156(4221)F.11[3,5]
30-27355(4441)H.4[5]
31-27640(1332)E.9[3,5]
32-27991(2122)H.4[5]
33-28031(4332)D.10[5]
34-28279(1432)G.16[5]
35-28759(2414)D.10[5]
---------------------------
36-31271(4112)E.14[7]
37-31639(1331)D.5[7]
38-31908(3211)F.12[3,7]
39-31999(1133)D.5[7]
40-32968(3232)D.5[3,7]
41-34027(3313)H.4.V1[7]
42-34088(4212)D.10[3,7]
43-34507(4334)D.5[7]
44-34867(1134)E.8[7]
45-35367(4224)D.5[7]
46-35539(4231)G.16[7]
47-36276(4442)H.4[3,7]
48-36807(4212)D.10[7]
49-37219(3343)H.4[7]
50-37540(4442)H.4[3,7]
51-37988(3231)E.9[3,7]
52-39736(4134)E.9[3,7]
53-39819(3343)H.4[7]
---------------------------
54-40299(2241)D.10[7]
55-40692(2414)D.10[3,7]
56-41015(1312)D.10[7]
57-41063(2422)H.4[7]
58-41583(1221)D.5[7]
59-41671(1331)D.5[7]
60-42423(4332)D.10[7]
61-42619(3234)E.8[7]
62-42859(4313)E.14[7]
63-43192(2213)D.10[3,7]
64-43307(2244)D.5[7]
65-43827(3313)H.4[7]
66-43847(2311)E.14[7]
67-44004(3431)D.10[3,7]
68-45835(2414)D.10[7]
69-45887(3244)E.9[7]
70-46551(2122)H.4[7]
71-46587(4223)D.10[7]
72-48052(1312)D.10[3,7]
73-48472(4134)E.9[3,7]
74-48667(2231)E.9[7]
75-49128(4133)D.10[3,7]
76-49812(3242)D.10[3,7]
77-49924(3412)G.19[3,7]
---------------------------
78-50739(3144)D.10[11]
79-50855(2414)D.10[11]
80-50983(4134)E.9[11]
81-51348(2313)E.14[3,11]
82-51995(1413)D.10[11]
83-53839(2111)H.4[11]
84-53843(4142)E.14[11]
85-54071(4111)H.4[11]
86-54195(3412)G.19[11]
87-54251(4441)H.4[11]
88-54319(4234)E.8[11]
89-55247(2241)D.10[11]
90-55271D.10
91-55623D.10
92-57079D.5
93-58920(4134)E.9[3]
---------------------------
94-60099G.19
95-60196(4214)E.9[3]
96-60895(3344)E.6
97-63079(2111)H.4[11]
98-63103(2434)E.9
99-63303(3122)E.14
No.dK(k(1)…k(4))TypeRef.
100-64196(3434)D.5[3]
101-64952(4111)H.4.V1[3]
102-65051(4214)E.9
103-65203H.4.V1
104-65204(4114)E.6[3]
105-65407(3423)E.14
106-67480(2343)F.13[3,8]
107-68584(4122)E.14[3]
---------------------------
108-70244(4212)D.10[3]
109-72435D.10
110-72591H.4.V1
111-73007H.4
112-73448(1231)E.8[3]
113-73731D.5
114-75847H.4
115-77144(3221)D.10[3]
116-78180(2411)E.14[3]
117-78708(1421)D.10[3]
118-79163(2421)E.14
---------------------------
119-80516(2231)E.9[3]
120-81867D.10
121-83395G.19
122-84072(3111)H.4[3]
123-85199D.10
124-85796(1414)D.5[3]
125-86551G.19
126-87503D.10
127-87720(1221)D.5[3]
128-87727D.10
129-88447D.10
130-89924(3341)E.14[3]
---------------------------
131-90163(4413)E.14
132-91471D.10
133-91643G.19
134-91860(2433)D.10[3]
135-92660(1324)G.16[3]
136-92712(1421)D.10[3]
137-92827G.16
138-93067G.19
139-93207(3423)E.14
140-93823D.5
141-94420(1142)D.10[3]
142-95155D.10
143-95448(3441)E.14[3]
144-95691D.5
145-96436H.4
146-96551G.19
147-96827(2143)G.19.V1[8]
148-97063H.4
149-97555D.10
150-97583H.4
151-97687(1431)E.9
152-97799H.4
153-98347H.4
154-98795D.10
155-99707D.10
156-99939(4133)D.10[11]
---------------------------
166-104627(3442)F.13[8]
210-124363(4411)F.7[8]
220-128451(1214)E.8[11]
235-135059(1243)G.16.V1[8]
236-135587(1422)F.12[8]
261-156452(4321)G.19.V1[8]
268-159208(2343)F.13.V1[8]
271-160403(3314)F.12[8]
288-167064(2343)F.13[8]
314-184132(4211)F.12[8]
317-185747(1243)G.16.V1[8]
320-186483(3343)H.4.V2[8]
324-187503(2231)E.9[11]
330-189959(1323)F.12[8]
349-199735(2143)G.19.V2[10]
---------------------------
383-216987(4134)E.9[11]
399-224580(2443)F.13[10]
401-225299(3443)F.7[10]
430-241160(1143)F.11[10]
439-245463(1324)G.16.V1[10]
446-249371(4243)F.12.V1[10]
453-256935(4443)H.4.V3[10]
459-260515(3443)F.7[10]
462-262628(2343)F.13.V1[10]
463-262744(2441)E.14.V1[10]
465-263908(2313)E.14.V1[10]
471-268040(1441)E.6.V1[10]
480-273284(2412)F.13.V1[10]
490-278427(1443)F.12.V1[10]
500-283908(1413)D.10[11]
504-287155(3143)F.13[10]
514-290703(1324)G.16.V2[10]
515-291220(1443)F.12[10]
526-296407(2443)F.13[10]
528-297079(1431)E.9.V1[10]
---------------------------
617-344667(2443)F.13[11]
856-453423(4214)E.9[11]
875-461847(2344)D.10[11]
978-516756(1341)D.10[11]
1189-620328(2111)H.4.V4[11]
1215-629295(1414)D.5[11]
1255-642084(4122)E.14.V1[11]

2.2. 3-Principalization over Real Quadratic Fields.

In the following tables, we give the natural principalization type (k(1)…k(4))
of real quadratic fields K = Q(d1/2) with discriminant 0 < d < 106
and 3-class group of type (3,3).

The entire range 0 < d < 106 is covered completely now.
Our computations of 2009 for 119 new cases in the range 300000 < d < 1000000 [11]
and of 2006 for the 14 discriminants in the range 200000 < d < 300000 [10]
extend our own results for 11 fields with 100000 < d < 200000 of 1991 [6]
and the 5 examples with 0 < d < 100000 of Heider and Schmithals in 1982 [2] .

Rare or exceptional cases are printed in boldface digits.
For the meaning of an asterisk (*) see the explanations for d = 142097.

No.dK(k(1)…k(4))TypeRef.
132009(0003)a.3[2]
242817(0003)a.3[2]
362501(0000)a.1[2]
472329(0200)a.2[2]
594636(0030)a.2[2]
---------------------------
6103809(0003)a.3[6]
7114889(0020)a.3[6]
8130397(0003)a.3[6]
9142097(4000)a.3*[6]
10151141(0300)a.3[6]
11152949(0000)a.1[6]
12153949(1000)a.2[6]
13172252(0100)a.3[6]
14173944(0400)a.3*[6]
15184137(0300)a.3[6]
16189237(1000)a.2[6]
---------------------------
17206776(0200)a.2[10]
18209765(0200)a.2[10]
19213913(0400)a.3[10]
20214028(0030)a.2[10]
21214712(4321)G.19[10]
22219461(0030)a.2[10]
23220217(0003)a.3[10]
24250748(3000)a.3[10]
25252977(0000)a.1[10]
26259653(0100)a.3*[10]
27265245(0040)a.3[10]
28275881(0030)a.2[10]
29283673(0010)a.3*[10]
30298849(0001)a.3[10]
---------------------------
31320785(0100)a.3*[11]
32321053(0400)a.3*[11]
33326945(0100)a.3*[11]
34333656(0003)a.3[11]
35335229(3000)a.3*[11]
36341724(0002)a.3[11]
37342664(4134)E.9[11]
38358285(0000)a.1[11]
39363397(0020)a.3[11]
40371965(0003)a.3[11]
41390876(1000)a.2[11]
---------------------------
42400369(0004)a.2[11]
43412277(0003)a.3*[11]
44415432(0020)a.3[11]
45422573(1341)D.10[11]
46424236(0010)a.3*[11]
47431761(0030)a.2[11]
48449797(0010)a.3[11]
49459964(0010)a.3*[11]
50460817(1000)a.2[11]
51468472(3000)a.3[11]
52471057(3000)a.3[11]
53471713(0020)a.3*[11]
54476124(0400)a.3[11]
55476152(0040)a.3*[11]
56486221(0003)a.3[11]
57486581(1000)a.2[11]
58494236(3000)a.3.V1[11]
---------------------------
59502796(4332)D.10[11]
60510337(0300)a.3[11]
61527068(0020)a.3*[11]
62531437(0000)a.1[11]
63531445(0003)a.3[11]
64534824(0313)c.18[11]
65535441(2000)a.3*[11]
66540365(0231)c.21[11]
67548549(0004)a.2[11]
68549133(0002)a.3[11]
69551384(4000)a.3*[11]
70551692(0004)a.2[11]
71552392(0200)a.2[11]
72557657(0020)a.3[11]
73567473(3000)a.3*[11]
74575729(3221)D.10[11]
No.dK(k(1)…k(4))TypeRef.
75578581(0400)a.3[11]
76586760(0000)a.1[11]
77593941(0040)a.3[11]
78595009(0000)a.1[11]
79597068(3000)a.3[11]
---------------------------
80600085(3000)a.3[11]
81602521(0004)a.2[11]
82621429(4000)a.3[11]
83621749(0020)a.3*[11]
84626411(4223)D.10[11]
85631769(3434)D.5[11]
86636632(0001)a.3[11]
87637820(0400)a.3*[11]
88654796(0400)a.3[11]
89665832(0040)a.3[11]
90681276(0020)a.3*[11]
91686977(0010)a.3*[11]
92689896(0300)a.3[11]
93698556(1000)a.2[11]
---------------------------
94710652(0043)b.10[11]
95718705(0003)a.3[11]
96719105(0100)a.3[11]
97722893(0002)a.3[11]
98726933(0000)a.1[11]
99729293(0020)a.3*[11]
100747496(0020)a.3*[11]
101750376(2000)a.3*[11]
102751657(0003)a.3[11]
103775480(0004)a.2[11]
104775661(0030)a.2[11]
105781177(0200)a.2[11]
106782737(0100)a.3*[11]
107782876(1000)a.2[11]
108784997(0001)a.3*[11]
109785269(3000)a.3[11]
110790085(1000)a.2.V1[11]
---------------------------
111801368(0000)a.1[11]
112804648(0004)a.2[11]
113807937(0010)a.3*[11]
114810661(4223)D.10[11]
115814021(0040)a.3[11]
116823512(0040)a.3[11]
117829813(2000)a.3[11]
118831484(1000)a.2[11]
119835853(1144)D.5[11]
120836493(0030)a.2[11]
121859064(4224)D.5[11]
122873969(0040)a.3[11]
123874684(2000)a.3[11]
124881689(0300)a.3[11]
125893029(0100)a.3*[11]
126893689(1000)a.2[11]
---------------------------
127902333(0001)a.3[11]
128907629(0030)a.2[11]
129907709(0030)a.2[11]
130908241(0003)a.3[11]
131916181(0400)a.3*[11]
132935665(0010)a.3[11]
133939569(0400)a.3[11]
134940593(0000)a.1[11]
135942961(3000)a.3*[11]
136943077(4321)G.19[11]
137944760(0040)a.3[11]
138945813(0231)c.21[11]
139957013(2122)H.4[11]
140957484(0200)a.2[11]
141959629(0004)a.2[11]
142966053(4000)a.3*[11]
143966489(0000)a.1[11]
144967928(0020)a.3*[11]
145974157(0001)a.3*[11]
146980108(0004)a.2[11]
147982049(0010)a.3*[11]
148993349(0004)a.2[11]
149994008(1000)a.2[11]

3. Minimal Occurrences of the Principalization Types.

The principalization type (k(1)…k(4))
with respect to the four unramified cyclic cubic extensions N1,…,N4
of a quadratic number field K with 3-class group Syl3(C(K)) of type (3,3)
together with the family of 3-class numbers (h1,…,h4) of the absolute cubic subfields L1,…,L4
of the normal S3-fields N1,…,N4 between K1 and K
uniquely determines the class m-1 and order 3n of the 2-stage metabelian 3-group G = Gal(K2 | K)
of the 2nd Hilbert 3-class field K2 over the quadratic field K.

In the following table, the principalization types (k(1)…k(4)) are arranged into sections,
according to [1,9] , and they are numbered similarly as in [5,9] .
We always give a canonical representative (CR) of the type's equivalence class (S4-orbit),
the number of fixed points (FP),
the occupation numbers (ON) (telling how often each of the digits 0,1,2,3,4 appears in the representative),
the cardinality (#) of the type's orbit under the operation of S4,
the year of the concrete numerical realization of the type with a reference,
the smallest absolute discriminant |dK| of a quadratic field K with that type,
an ideal of polynomials in Z[X,Y], called the associated symbolic order (SO) in [1,3,9] ,
the structure of the commutator subgroup G' = G2,
the exponents in defining relations for the group's generators (RE),
and the set CBF(m,n), defined in [9] , to which the group G belongs.

Remarks: 1. Section "A" is impossible for quadratic base fields K.
(However, it can occur for cyclic cubic base fields K [0].)
2. Sections "B", "C", and "e" cannot occur at all, for group theoretic reasons.
3. Sections "a", "b", "c", and "d" can occur only for real quadratic base fields K.

Sec.No.CRFPON#YeardKSOh1,…,h4G2REG in
A1111110400041931 [0]-L-(3)1,0;0CF 1a(3)
D
51212202200121981 [2]-12131L23,3,3,3(3,3,3)1,1,-1,1;0CBF 1a(4,5)
2009 [11]631769L23,3,3,3(3,3,3)1,1,-1,1;0CBF 1a(4,5)
10112310211024
1933 [1]-4027L23,3,3,3(3,3,3)0,0,-1,1;0CBF 1a(4,5)
2006 [11]422573L23,3,3,3(3,3,3)0,0,-1,1;0CBF 1a(4,5)
E
6112210220012
1981 [2]-15544X49,3,3,3(9,9,3)1,-1,1,1;0CBF 2a(6,7)
2005 [10]-268040X627,3,3,3(27,27,3)1,-1,1,1;0CBF 2a(8,9)
20105264069X49,3,3,3(9,9,3)1,-1,1,1;0CBF 2a(6,7)
8123130211012
2003 [7]-34867X49,3,3,3(9,9,3)1,0,-1,1;0CBF 2a(6,7)
2010-370740X627,3,3,3(27,27,3)1,0,-1,1;0CBF 2a(8,9)
20106098360X49,3,3,3(9,9,3)1,0,-1,1;0CBF 2a(6,7)
9121320211024
1933 [1]-9748X49,3,3,3(9,9,3)0,0,+-1,1;0CBF 2a(6,7)
2005 [10]-297079X627,3,3,3(27,27,3)0,0,+-1,1;0CBF 2a(8,9)
2006 [11]342664X49,3,3,3(9,9,3)0,0,+-1,1;0CBF 2a(6,7)
14231100211024
1981 [2]-16627X49,3,3,3(9,9,3)0,-1,+-1,1;0CBF 2a(6,7)
2005 [10]-262744X627,3,3,3(27,27,3)0,-1,+-1,1;0CBF 2a(8,9)
20103918837X49,3,3,3(9,9,3)0,-1,+-1,1;0CBF 2a(6,7)
F
7211200220012
2003 [8]-124363R4,49,9,3,3(9,9,9,3)+-1,1,+-1,1;0CBF 1a(6,9)
2010-469816R6,427,9,3,3(27,27,9,3)+-1,1,+-1,1;0CBF 2a(8,11)
11132110211012
1984 [3]-27156R4,49,9,3,3(9,9,9,3)1,+-1,0,0;0CBF 1a(6,9)
2010-469787R6,427,9,3,3(27,27,9,3)1,+-1,0,0;0CBF 2a(8,11)
12321110211024
1984 [3]-31908R4,49,9,3,3(9,9,9,3)+-1,+-1,0,+-1;0CBF 1a(6,9)
2005 [10]-249371R6,427,9,3,3(27,27,9,3)+-1,?,?,+-1;0CBF 2a(8,11)
2010-423640R6,627,27,3,3(27,27,27,9)+-1,?,?,+-1;0CBF 1a(8,13)
13211300211024
1984 [3]-67480R4,49,9,3,3(9,9,9,3)+-1,+-1,+-1,0;0CBF 1a(6,9)
2003 [8]-159208R6,427,9,3,3(27,27,9,3)?,+-1,+-1,?;0CBF 2a(8,11)
20108321505R4,49,9,3,3(9,9,9,3)+-1,+-1,+-1,0;0CBF 1a(6,9)
20108127208R6,427,9,3,3(27,27,9,3)?,+-1,+-1,?;0CBF 2a(8,11)
G
1621342011116
1981 [2]-17131Z59,3,3,3(27,9,3)?,?,?,?;1CBF 2b(7,8)
2010-819743Z727,3,3,3(81,27,3)?,?,?,?;1CBF 2b(9,10)
2003 [8]-135059V5,59,9,3,3(9,9,9,9)+-1,0,+-1,1;-1CBF 1b(7,10)
2006 [10]-290703V'5,59,9,3,3(27,9,9,3)?,0,?,+-1;1CBF 1b(7,10)
20108711453Z59,3,3,3(27,9,3)?,?,?,?;1CBF 2b(7,8)
1921430011113
1981 [2]-12067Z3,3,3,3(3,3,3,3)+-1,0,1,-1;-1CBF 1b(5,6)
2003 [8]-96827T'5,59,9,3,3(27,9,9,3)?,+-1,?,0;1CBF 1b(7,10)
2005 [10]-199735T5,59,9,3,3(9,9,9,9)?,+-1,?,0;-1CBF 1b(7,10)
2010-509160T'7,527,9,3,3(81,27,9,3)?,+-1,?,0;1CBF 2b(9,12)
2006 [10]214712Z3,3,3,3(3,3,3,3)+-1,0,1,-1;-1CBF 1b(5,6)
H4211100310012
1981 [2]-3896Z'3,3,3,3(9,3,3)+-1,1,+-1,+-1;1CBF 1b(5,6)
2003 [3,5,7]-21668Z'59,3,3,3(27,9,3)+-1,-1,+-1,+-1;-1CBF 2b(7,8)
2009 [11]-446788Z'727,3,3,3(81,27,3)+-1,-1,+-1,+-1;-1CBF 2b(9,10)
2004 [8]-186483V5,59,9,3,3(9,9,9,9)+-1,1,+-1,+-1;-1CBF 1b(7,10)
2006 [10]-256935V'5,59,9,3,3(27,9,9,3)?,+-1,?,+-1;1CBF 1b(7,10)
2010-678804V7,527,9,3,3(81,27,9,3)+-1,1,+-1,+-1;-1CBF 2b(9,12)
2009 [11]957013Z'3,3,3,3(9,3,3)+-1,1,+-1,+-1;1CBF 1b(5,6)
20101162949Z'59,3,3,3(27,9,3)+-1,-1,+-1,+-1;-1CBF 2b(7,8)
a
100000400001
1981 [2]62501Y'49,3,3,3(9,9)?,0;1CF 2b(6)
20102905160Y'627,3,3,3(27,27)?,0;1CF 2b(8)
210001310004
1981 [2]72329Y23,3,3,3(3,3)1,0;0CF 2a(4)
2009 [11]790085Y49,3,3,3(9,9)1,0;0CF 2a(6)
3010003100012
1981 [2]32009Y23,3,3,3(3,3)0,-1;0CF 2a(4)
2009 [11]142097Y23,3,3,3(3,3)0,+1;0CF 2a(4)
2008 [11]494236Y49,3,3,3(9,9)0,+-1;0CF 2a(6)
b
10210002110062009 [11]710652V4,49,9,3,3(9,9,3,3)0,0,0,0;1CBF 1b(6,8)
c
182011012100122009 [11]534824X39,3,3,3(9,3,3)0,-1,0,1;0CBF 2a(5,6)
21120321111012
2008 [11]540365X39,3,3,3(9,3,3)0,0,0,1;0CBF 2a(5,6)
20101001957X527,3,3,3(27,9,3)0,0,0,1;0CBF 2a(7,8)
d
1921100121002420102328721R4,39,9,3,3(9,9,3,3)1,0,+-1,0;0CBF 2a(6,8)
2313201111101220101535117R4,39,9,3,3(9,9,3,3)1,0,0,0;0CBF 2a(6,8)
25*03210111101220108491713R5,427,9,3,3(27,9,9,3)0,+-1,0,0;0CBF 2a(7,10)


Bibliographic References:

[0] Olga Taussky,
Über eine Verschärfung des Hauptidealsatzes
für algebraische Zahlkörper,
J. reine angew. Math. 168 (1932), 193 - 210

[1] Arnold Scholz und Olga Taussky,
Die Hauptideale der kubischen Klassenkörper
imaginär quadratischer Zahlkörper,
J. reine angew. Math. 171 (1934), 19 - 41

[2] Franz-Peter Heider und Bodo Schmithals,
Zur Kapitulation der Idealklassen
in unverzweigten primzyklischen Erweiterungen,
J. reine angew. Math. 336 (1982), 1 - 25

[3] James R. Brink,
The class field tower for imaginary quadratic number fields of type (3,3),
Dissertation, Ohio State Univ., 1984.

[4] James R. Brink and Robert Gold,
Class field towers of imaginary quadratic fields,
manuscripta math. 57 (1987), 425 - 450

[5] Daniel C. Mayer,
Principalization in complex S3-fields,
Congressus Numerantium 80 (1991), 73 - 87

[6] Daniel C. Mayer,
List of discriminants dL < 200000 of totally real cubic fields L,
arranged according to their multiplicities m and conductors f,
1991, Dept. of Comp. Sci., Univ. of Manitoba

[7] Daniel C. Mayer,
Principalization in Unramified Cyclic Cubic Extensions
of all Quadratic Fields with Discriminant -50000 < d < 0
and 3-Class Group of Type (3,3),
Univ. Graz, Computer Centre, 2003

[8] Daniel C. Mayer,
Principalization in Unramified Cyclic Cubic Extensions
of selected Quadratic Fields with Discriminant -200000 < d < -50000
and 3-Class Group of Type (3,3),
Univ. Graz, Computer Centre, 2004

[9] Daniel C. Mayer,
Two-Stage Towers of 3-Class Fields over Quadratic Fields,
(Latest Update)
Univ. Graz, 2009.

[10] Daniel C. Mayer,
3-Capitulation over Quadratic Fields
with Discriminant |d| < 3*105 and 3-Class Group of Type (3,3),
Univ. Graz, Computer Centre, 2006.

[11] Daniel C. Mayer,
3-Capitulation over Quadratic Fields
with Discriminant |d| < 106 and 3-Class Group of Type (3,3),
(Latest Update)
Univ. Graz, Computer Centre, 2009.

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