All known Examples for Principalization Types.

1. Definition of the Principalization Type.

Assume that p denotes an odd rational prime.

Let K be a base field with p-class rank 2 and
with p-class group Syl_p(C(K)) of p-elementary abelian type (p,p).
Denote by K₁ the 1^st Hilbert p-class field of K.
According to the Artin Reciprocity Law of Class Field Theory,
the p-class group Syl_p(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 C_i (1 <= i <= p+1) of Syl_p(C(K)) are mapped to the
Galois groups Gal(K₁|N_i) = M_i of p+1 unramified cyclic extensions N_i of K of p-th degree.
In fact, the C_i = Norm(N_i|K)Syl_p(C(N_i)) are relative norms of p-class groups.
The following diagrams illustrate the Galois correspondence and the Artin isomorphism:

				K₁ = N₀
	/		/		\		\
N₁		N₂				...		N_p+1
	\		\		/		/
				K

		Gal(K₁\|K₁) = 1
	/	\|	\
M₁		Gal(K₁\|N_i) = M_i		M_p+1
	\	\|	/
		Gal(K₁\|K)

				1
	/		/		\		\
C₁		C₂				...		C_p+1
	\		\		/		/
				Syl_p(C(K)) = C₀

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

According to Hilbert's Theorem 94, none of the principalization kernels Kern j(N_i|K) is trivial,
since there is an isomorphism to the unit cohomology of N = N_i:
Kern j(N|K) = P_N^<S>/P_K = E_N|K/U_N^S-1 <> 1,
where Gal(N|K) = <S> and E_N|K denotes the intersection of U_N 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) = Syl_p(C(K)) = C₀.

Thus, there are p+2 possibilities for each Kern j(N_i|K), C₀,C₁,...,C_p+1,
and if K is a quadratic field we define a Natural Principalization Type (k(1),...,k(p+1)),
ordering the N_i (1 <= i <= p+1), which have dihedral absolute groups Gal(N_i|Q) = D(2p),
by increasing regulators of their absolute subfields L_i 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(N_i|K) = C_k(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(d^1/2) 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.	d_K	(k(1)...k(4))	Type	Ref.
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]

No.	d_K	(k(1)...k(4))	Type	Ref.
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]
93	-58920	(4134)	E.9	[3]
---	-------	------	------	-----
95	-60196	(4214)	E.9	[3]
97	-63079	(2111)	H.4	[11]
100	-64196	(3434)	D.5	[3]
101	-64952	(4111)	H.4.V1	[3]
104	-65204	(4114)	E.6	[3]
106	-67480	(2343)	F.13	[3,8]
107	-68584	(4122)	E.14	[3]
---	-------	------	------	-----
108	-70244	(4212)	D.10	[3]
112	-73448	(1231)	E.8	[3]
115	-77144	(3221)	D.10	[3]
116	-78180	(2411)	E.14	[3]
117	-78708	(1421)	D.10	[3]
---	-------	------	------	-----
119	-80516	(2231)	E.9	[3]
122	-84072	(3111)	H.4	[3]
124	-85796	(1414)	D.5	[3]
127	-87720	(1221)	D.5	[3]
130	-89924	(3341)	E.14	[3]
---	-------	------	------	-----
134	-91860	(2433)	D.10	[3]
135	-92660	(1324)	G.16	[3]
136	-92712	(1421)	D.10	[3]
141	-94420	(1142)	D.10	[3]
143	-95448	(3441)	E.14	[3]
147	-96827	(2143)	G.19.V1	[8]
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(d^1/2) 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.	d_K	(k(1)...k(4))	Type	Ref.
1	32009	(0003)	a.3	[2]
2	42817	(0003)	a.3	[2]
3	62501	(0000)	a.1	[2]
4	72329	(0200)	a.2	[2]
5	94636	(0030)	a.2	[2]
---	-------	------	------	-----
6	103809	(0003)	a.3	[6]
7	114889	(0020)	a.3	[6]
8	130397	(0003)	a.3	[6]
9	142097	(4000)	a.3^*	[6]
10	151141	(0300)	a.3	[6]
11	152949	(0000)	a.1	[6]
12	153949	(1000)	a.2	[6]
13	172252	(0100)	a.3	[6]
14	173944	(0400)	a.3^*	[6]
15	184137	(0300)	a.3	[6]
16	189237	(1000)	a.2	[6]
---	-------	------	------	-----
17	206776	(0200)	a.2	[10]
18	209765	(0200)	a.2	[10]
19	213913	(0400)	a.3	[10]
20	214028	(0030)	a.2	[10]
21	214712	(4321)	G.19	[10]
22	219461	(0030)	a.2	[10]
23	220217	(0003)	a.3	[10]
24	250748	(3000)	a.3	[10]
25	252977	(0000)	a.1	[10]
26	259653	(0100)	a.3^*	[10]
27	265245	(0040)	a.3	[10]
28	275881	(0030)	a.2	[10]
29	283673	(0010)	a.3^*	[10]
30	298849	(0001)	a.3	[10]
---	-------	------	------	-----
31	320785	(0100)	a.3^*	[11]
32	321053	(0400)	a.3^*	[11]
33	326945	(0100)	a.3^*	[11]
34	333656	(0003)	a.3	[11]
35	335229	(3000)	a.3^*	[11]
36	341724	(0002)	a.3	[11]
37	342664	(4134)	E.9	[11]
38	358285	(0000)	a.1	[11]
39	363397	(0020)	a.3	[11]
40	371965	(0003)	a.3	[11]
41	390876	(1000)	a.2	[11]
---	-------	------	------	-----
42	400369	(0004)	a.2	[11]
43	412277	(0003)	a.3^*	[11]
44	415432	(0020)	a.3	[11]
45	422573	(1341)	D.10	[11]
46	424236	(0010)	a.3^*	[11]
47	431761	(0030)	a.2	[11]
48	449797	(0010)	a.3	[11]
49	459964	(0010)	a.3^*	[11]
50	460817	(1000)	a.2	[11]
51	468472	(3000)	a.3	[11]
52	471057	(3000)	a.3	[11]
53	471713	(0020)	a.3^*	[11]
54	476124	(0400)	a.3	[11]
55	476152	(0040)	a.3^*	[11]
56	486221	(0003)	a.3	[11]
57	486581	(1000)	a.2	[11]
58	494236	(3000)	a.3.V1	[11]
---	-------	------	------	-----
59	502796	(4332)	D.10	[11]
60	510337	(0300)	a.3	[11]
61	527068	(0020)	a.3^*	[11]
62	531437	(0000)	a.1	[11]
63	531445	(0003)	a.3	[11]
64	534824	(0313)	c.18	[11]
65	535441	(2000)	a.3^*	[11]
66	540365	(0231)	c.21	[11]
67	548549	(0004)	a.2	[11]
68	549133	(0002)	a.3	[11]
69	551384	(4000)	a.3^*	[11]
70	551692	(0004)	a.2	[11]
71	552392	(0200)	a.2	[11]
72	557657	(0020)	a.3	[11]
73	567473	(3000)	a.3^*	[11]
74	575729	(3221)	D.10	[11]

No.	d_K	(k(1)...k(4))	Type	Ref.
75	578581	(0400)	a.3	[11]
76	586760	(0000)	a.1	[11]
77	593941	(0040)	a.3	[11]
78	595009	(0000)	a.1	[11]
79	597068	(3000)	a.3	[11]
---	-------	------	------	-----
80	600085	(3000)	a.3	[11]
81	602521	(0004)	a.2	[11]
82	621429	(4000)	a.3	[11]
83	621749	(0020)	a.3^*	[11]
84	626411	(4223)	D.10	[11]
85	631769	(3434)	D.5	[11]
86	636632	(0001)	a.3	[11]
87	637820	(0400)	a.3^*	[11]
88	654796	(0400)	a.3	[11]
89	665832	(0040)	a.3	[11]
90	681276	(0020)	a.3^*	[11]
91	686977	(0010)	a.3^*	[11]
92	689896	(0300)	a.3	[11]
93	698556	(1000)	a.2	[11]
---	-------	------	------	-----
94	710652	(0043)	b.10	[11]
95	718705	(0003)	a.3	[11]
96	719105	(0100)	a.3	[11]
97	722893	(0002)	a.3	[11]
98	726933	(0000)	a.1	[11]
99	729293	(0020)	a.3^*	[11]
100	747496	(0020)	a.3^*	[11]
101	750376	(2000)	a.3^*	[11]
102	751657	(0003)	a.3	[11]
103	775480	(0004)	a.2	[11]
104	775661	(0030)	a.2	[11]
105	781177	(0200)	a.2	[11]
106	782737	(0100)	a.3^*	[11]
107	782876	(1000)	a.2	[11]
108	784997	(0001)	a.3^*	[11]
109	785269	(3000)	a.3	[11]
110	790085	(1000)	a.2.V1	[11]
---	-------	------	------	-----
111	801368	(0000)	a.1	[11]
112	804648	(0004)	a.2	[11]
113	807937	(0010)	a.3^*	[11]
114	810661	(4223)	D.10	[11]
115	814021	(0040)	a.3	[11]
116	823512	(0040)	a.3	[11]
117	829813	(2000)	a.3	[11]
118	831484	(1000)	a.2	[11]
119	835853	(1144)	D.5	[11]
120	836493	(0030)	a.2	[11]
121	859064	(4224)	D.5	[11]
122	873969	(0040)	a.3	[11]
123	874684	(2000)	a.3	[11]
124	881689	(0300)	a.3	[11]
125	893029	(0100)	a.3^*	[11]
126	893689	(1000)	a.2	[11]
---	-------	------	------	-----
127	902333	(0001)	a.3	[11]
128	907629	(0030)	a.2	[11]
129	907709	(0030)	a.2	[11]
130	908241	(0003)	a.3	[11]
131	916181	(0400)	a.3^*	[11]
132	935665	(0010)	a.3	[11]
133	939569	(0400)	a.3	[11]
134	940593	(0000)	a.1	[11]
135	942961	(3000)	a.3^*	[11]
136	943077	(4321)	G.19	[10]
137	944760	(0040)	a.3	[11]
138	945813	(0231)	c.21	[11]
139	957013	(2122)	H.4	[11]
140	957484	(0200)	a.2	[11]
141	959629	(0004)	a.2	[11]
142	966053	(4000)	a.3^*	[11]
143	966489	(0000)	a.1	[11]
144	967928	(0020)	a.3^*	[11]
145	974157	(0001)	a.3^*	[11]
146	980108	(0004)	a.2	[11]
147	982049	(0010)	a.3^*	[11]
148	993349	(0004)	a.2	[11]
149	994008	(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 2^nd 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 |d_K| 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.	CR	FP	ON	#	Year	d_K	SO	h₁,...,h₄	G₂	RE	G in
A	1	1111	1	04000	4	1931 [0]	-	L	-	(3)	1,0;0	CF 1a(3)
D
	5	1212	2	02200	12	1981 [2]	-12131	L₂	3,3,3,3	(3,3,3)	1,1,-1,1;0	CBF 1a(4,5)
						2009 [11]	631769	L₂	3,3,3,3	(3,3,3)	1,1,-1,1;0	CBF 1a(4,5)
	10	1123	1	02110	24
						1933 [1]	-4027	L₂	3,3,3,3	(3,3,3)	0,0,-1,1;0	CBF 1a(4,5)
						2006 [11]	422573	L₂	3,3,3,3	(3,3,3)	0,0,-1,1;0	CBF 1a(4,5)
E
	6	1122	1	02200	12
						1981 [2]	-15544	X₄	9,3,3,3	(9,9,3)	1,-1,1,1;0	CBF 2a(6,7)
						2005 [10]	-268040	X₆	27,3,3,3	(27,27,3)	1,-1,1,1;0	CBF 2a(8,9)
						2010	5264069	X₄	9,3,3,3	(9,9,3)	1,-1,1,1;0	CBF 2a(6,7)
	8	1231	3	02110	12
						2003 [7]	-34867	X₄	9,3,3,3	(9,9,3)	1,0,-1,1;0	CBF 2a(6,7)
						2010	-370740	X₆	27,3,3,3	(27,27,3)	1,0,-1,1;0	CBF 2a(8,9)
						2010	6098360	X₄	9,3,3,3	(9,9,3)	1,0,-1,1;0	CBF 2a(6,7)
	9	1213	2	02110	24
						1933 [1]	-9748	X₄	9,3,3,3	(9,9,3)	0,0,+-1,1;0	CBF 2a(6,7)
						2005 [10]	-297079	X₆	27,3,3,3	(27,27,3)	0,0,+-1,1;0	CBF 2a(8,9)
						2006 [11]	342664	X₄	9,3,3,3	(9,9,3)	0,0,+-1,1;0	CBF 2a(6,7)
	14	2311	0	02110	24
						1981 [2]	-16627	X₄	9,3,3,3	(9,9,3)	0,-1,+-1,1;0	CBF 2a(6,7)
						2005 [10]	-262744	X₆	27,3,3,3	(27,27,3)	0,-1,+-1,1;0	CBF 2a(8,9)
						2010	3918837	X₄	9,3,3,3	(9,9,3)	0,-1,+-1,1;0	CBF 2a(6,7)
F
	7	2112	0	02200	12
						2003 [8]	-124363	R_4,4	9,9,3,3	(9,9,9,3)	+-1,1,+-1,1;0	CBF 1a(6,9)
						2010	-469816	R_6,4	27,9,3,3	(27,27,9,3)	+-1,1,+-1,1;0	CBF 2a(8,11)
	11	1321	1	02110	12
						1984 [3]	-27156	R_4,4	9,9,3,3	(9,9,9,3)	1,+-1,0,0;0	CBF 1a(6,9)
						2010	-469787	R_6,4	27,9,3,3	(27,27,9,3)	1,+-1,0,0;0	CBF 2a(8,11)
	12	3211	1	02110	24
						1984 [3]	-31908	R_4,4	9,9,3,3	(9,9,9,3)	+-1,+-1,0,+-1;0	CBF 1a(6,9)
						2005 [10]	-249371	R_6,4	27,9,3,3	(27,27,9,3)	+-1,?,?,+-1;0	CBF 2a(8,11)
						2010	-423640	R_6,6	27,27,3,3	(27,27,27,9)	+-1,?,?,+-1;0	CBF 1a(8,13)
	13	2113	0	02110	24
						1984 [3]	-67480	R_4,4	9,9,3,3	(9,9,9,3)	+-1,+-1,+-1,0;0	CBF 1a(6,9)
						2003 [8]	-159208	R_6,4	27,9,3,3	(27,27,9,3)	?,+-1,+-1,?;0	CBF 2a(8,11)
						2010	8321505	R_4,4	9,9,3,3	(9,9,9,3)	+-1,+-1,+-1,0;0	CBF 1a(6,9)
						2010	8127208	R_6,4	27,9,3,3	(27,27,9,3)	?,+-1,+-1,?;0	CBF 2a(8,11)
G
	16	2134	2	01111	6
						1981 [2]	-17131	Z₅	9,3,3,3	(27,9,3)	?,?,?,?;1	CBF 2b(7,8)
						2010	-819743	Z₇	27,3,3,3	(81,27,3)	?,?,?,?;1	CBF 2b(9,10)
						2003 [8]	-135059	V_5,5	9,9,3,3	(9,9,9,9)	+-1,0,+-1,1;-1	CBF 1b(7,10)
						2006 [10]	-290703	V^'_5,5	9,9,3,3	(27,9,9,3)	?,0,?,+-1;1	CBF 1b(7,10)
						2010	8711453	Z₅	9,3,3,3	(27,9,3)	?,?,?,?;1	CBF 2b(7,8)
	19	2143	0	01111	3
						1981 [2]	-12067	Z	3,3,3,3	(3,3,3,3)	+-1,0,1,-1;-1	CBF 1b(5,6)
						2003 [8]	-96827	T^'_5,5	9,9,3,3	(27,9,9,3)	?,+-1,?,0;1	CBF 1b(7,10)
						2005 [10]	-199735	T_5,5	9,9,3,3	(9,9,9,9)	?,+-1,?,0;-1	CBF 1b(7,10)
						2010	-509160	T^'_7,5	27,9,3,3	(81,27,9,3)	?,+-1,?,0;1	CBF 2b(9,12)
						2006 [10]	214712	Z	3,3,3,3	(3,3,3,3)	+-1,0,1,-1;-1	CBF 1b(5,6)
H	4	2111	0	03100	12
						1981 [2]	-3896	Z^'	3,3,3,3	(9,3,3)	+-1,1,+-1,+-1;1	CBF 1b(5,6)
						2003 [3,5,7]	-21668	Z^'₅	9,3,3,3	(27,9,3)	+-1,-1,+-1,+-1;-1	CBF 2b(7,8)
						2009 [11]	-446788	Z^'₇	27,3,3,3	(81,27,3)	+-1,-1,+-1,+-1;-1	CBF 2b(9,10)
						2004 [8]	-186483	V_5,5	9,9,3,3	(9,9,9,9)	+-1,1,+-1,+-1;-1	CBF 1b(7,10)
						2006 [10]	-256935	V^'_5,5	9,9,3,3	(27,9,9,3)	?,+-1,?,+-1;1	CBF 1b(7,10)
						2010	-678804	V_7,5	27,9,3,3	(81,27,9,3)	+-1,1,+-1,+-1;-1	CBF 2b(9,12)
						2009 [11]	957013	Z^'	3,3,3,3	(9,3,3)	+-1,1,+-1,+-1;1	CBF 1b(5,6)
						2010	1162949	Z^'₅	9,3,3,3	(27,9,3)	+-1,-1,+-1,+-1;-1	CBF 2b(7,8)
a
	1	0000	0	40000	1
						1981 [2]	62501	Y^'₄	9,3,3,3	(9,9)	?,0;1	CF 2b(6)
						2010	2905160	Y^'₆	27,3,3,3	(27,27)	?,0;1	CF 2b(8)
	2	1000	1	31000	4
						1981 [2]	72329	Y₂	3,3,3,3	(3,3)	1,0;0	CF 2a(4)
						2009 [11]	790085	Y₄	9,3,3,3	(9,9)	1,0;0	CF 2a(6)
	3	0100	0	31000	12
						1981 [2]	32009	Y₂	3,3,3,3	(3,3)	0,-1;0	CF 2a(4)
						2009 [11]	142097	Y₂	3,3,3,3	(3,3)	0,+1;0	CF 2a(4)
						2008 [11]	494236	Y₄	9,3,3,3	(9,9)	0,+-1;0	CF 2a(6)
b
	10	2100	0	21100	6	2009 [11]	710652	V_4,4	9,9,3,3	(9,9,3,3)	0,0,0,0;1	CBF 1b(6,8)
c
	18	2011	0	12100	12	2009 [11]	534824	X₃	9,3,3,3	(9,3,3)	0,-1,0,1;0	CBF 2a(5,6)
	21	1203	2	11110	12
						2008 [11]	540365	X₃	9,3,3,3	(9,3,3)	0,0,0,1;0	CBF 2a(5,6)
						2010	1001957	X₅	27,3,3,3	(27,9,3)	0,0,0,1;0	CBF 2a(7,8)
d
	19	2110	0	12100	24	2010	2328721	R_4,3	9,9,3,3	(9,9,3,3)	1,0,+-1,0;0	CBF 2a(6,8)
	23	1320	1	11110	12	2010	1535117	R_4,3	9,9,3,3	(9,9,3,3)	1,0,0,0;0	CBF 2a(6,8)
	25*	0321	0	11110	12	2010	8491713	R_5,4	27,9,3,3	(27,9,9,3)	0,+-1,0,0;0	CBF 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 d_L < 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.

Web master's e-mail address:
contact@algebra.at

Navigation Center

Back to Algebra