MAGMA 2012



Triadic Quantum Class Groups

over Bicyclic Biquadratic Fields of Gauss-Dirichlet-Hilbert Type

3.0. Real Quadratic Base Objects Q(D1/2)

Again, we started our investigations by using a table of
all 870 discriminants D in the range 0 < D < 3*104
of real quadratic fields R = Q(D1/2)
having class number h(R) divisible by 3.
Since the smallest discriminant with 3-class rank two is D = 32009,
all 870 fields have a non-trivial cyclic 3-class group Cl3(R).
However, we point out that, due to the lack of a reflection theorem,
these 870 discriminants will have to be supplemented
by certain discriminants of real quadratic fields with 3-rank 0.

3.1. Reflection at Gauss' Cyclotomic Q((-1)1/2)

3.1.0. Dual Complex Quadratic Base Objects Q((-D)1/2)

The 3-class group Cl3(C)
of the complex quadratic field C = Q((-D)1/2)
with dual discriminant d(C) of d(R) = D may have any 3-rank 0,1,2,….

3.1.1. Bicyclic Biquadratic Gauss-Dirichlet Composita Q((-1)1/2,D1/2)

Composition of R and C yields a bicyclic biquadratic Gauss-Dirichlet-Hilbert field B = R*C,
which is also constructible by composition B = R*K or B = C*K
with Gauss' cyclotomic K = Q((-1)1/2)
having discriminant d(K) = -4 and trivial class group Cl(K) = 1.

According to theories developed by Dirichlet, Hilbert, Herglotz, Kuroda, and Kubota,
the discriminant of B is the product d(B) = d(R)*d(C)*d(K) = gcd(d(R),d(C))2*d(K)2
and the 3-class group of B, Cl3(B) = Cl3(R)*Cl3(C), is a direct product.

It turned out that the bicyclic biquadratic fields B = Q((-1)1/2,D1/2)
constructed by composing the 870 real quadratic fields R = Q(D1/2),
and the supplementary real quadratic fields of 3-rank 0,
with Gauss' cyclotomic K = Q((-1)1/2)
are partitioned into the following three sets,
according to the structure of their 3-class group Cl3(B):
there are 211 fields of type (3,3),
90 fields of type (9,3),
and more than 569 other fields, mainly of 3-class rank 1.
The 211 bicyclic biquadratic fields B of type (3,3), 0 < D < 3*104,
are given in the following table.
(Output format is Counter: d(R) Cl(R), d(C) Cl(C), d(B) Cl(B).)
1: dr=473 gr=[3], dc=-1892 gc=[6, 2], db=3579664 gb=[6, 3]
2: dr=985 gr=[6], dc=-3940 gc=[12, 2], db=15523600 gb=[12, 6]
3: dr=993 gr=[3], dc=-3972 gc=[6, 2], db=15776784 gb=[6, 3]
4: dr=1901 gr=[3], dc=-7604 gc=[42], db=57820816 gb=[21, 3]
5: dr=1937 gr=[6], dc=-7748 gc=[24, 2], db=60031504 gb=[24, 6]
6: dr=2213 gr=[3], dc=-8852 gc=[42], db=78357904 gb=[21, 3]
7: dr=2437 gr=[], dc=-9748 gc=[6, 3], db=95023504 gb=[3, 3]
8: dr=2713 gr=[3], dc=-10852 gc=[24], db=117765904 gb=[12, 3]
9: dr=2920 gr=[6, 2], dc=-2920 gc=[6, 2], db=34105600 gb=[6, 6, 2]
10: dr=2993 gr=[6], dc=-11972 gc=[12, 4], db=143328784 gb=[12, 6, 2]
11: dr=3305 gr=[6], dc=-13220 gc=[24, 2], db=174768400 gb=[12, 12]
12: dr=3356 gr=[3], dc=-839 gc=[33], db=11262736 gb=[33, 3]
13: dr=3592 gr=[6], dc=-3592 gc=[12], db=51609856 gb=[12, 6]
14: dr=3877 gr=[3], dc=-15508 gc=[30], db=240498064 gb=[15, 3]
15: dr=3896 gr=[], dc=-3896 gc=[12, 3], db=60715264 gb=[12, 3]
16: dr=3941 gr=[3], dc=-15764 gc=[30, 2], db=248503696 gb=[30, 3]
17: dr=3957 gr=[3], dc=-15828 gc=[12, 2], db=250525584 gb=[12, 3]
18: dr=4345 gr=[12], dc=-17380 gc=[12, 2, 2], db=302064400 gb=[24, 12]
19: dr=4360 gr=[6, 2], dc=-4360 gc=[6, 2], db=76038400 gb=[6, 6, 2]
20: dr=4764 gr=[6], dc=-1191 gc=[24], db=22695696 gb=[24, 3]
21: dr=5069 gr=[2], dc=-20276 gc=[12, 6], db=411116176 gb=[12, 6]
22: dr=5073 gr=[6], dc=-20292 gc=[12, 2, 2], db=411765264 gb=[12, 6, 2]
23: dr=5353 gr=[6], dc=-21412 gc=[24, 2], db=458473744 gb=[24, 6]
24: dr=5417 gr=[7], dc=-21668 gc=[24, 3], db=469502224 gb=[84, 3]
25: dr=5497 gr=[3], dc=-21988 gc=[12, 2], db=483472144 gb=[12, 3]
26: dr=5613 gr=[3], dc=-22452 gc=[24, 2], db=504092304 gb=[24, 3]
27: dr=5857 gr=[], dc=-23428 gc=[12, 3], db=548871184 gb=[6, 3]
28: dr=5912 gr=[3], dc=-5912 gc=[30], db=139806976 gb=[30, 3]
29: dr=6085 gr=[10], dc=-24340 gc=[6, 6], db=592435600 gb=[30, 6]
30: dr=6133 gr=[3], dc=-24532 gc=[42], db=601819024 gb=[21, 3]
31: dr=6209 gr=[3], dc=-24836 gc=[48, 2], db=616826896 gb=[48, 3]
32: dr=6221 gr=[], dc=-24884 gc=[42, 3], db=619213456 gb=[21, 3]
33: dr=6268 gr=[3], dc=-1567 gc=[15], db=39287824 gb=[15, 3]
34: dr=6401 gr=[12], dc=-25604 gc=[60, 2], db=655564816 gb=[120, 6]
35: dr=6557 gr=[3], dc=-26228 gc=[24, 2], db=687907984 gb=[24, 3]
36: dr=6601 gr=[6], dc=-26404 gc=[12, 2, 2], db=697171216 gb=[12, 6, 2]
37: dr=6789 gr=[2], dc=-27156 gc=[6, 6, 2], db=737448336 gb=[6, 6, 2]
38: dr=7665 gr=[6, 2], dc=-30660 gc=[6, 2, 2, 2], db=940035600 gb=[12, 6, 2, 2]
39: dr=7881 gr=[12], dc=-31524 gc=[24, 2, 2], db=993762576 gb=[48, 12]
40: dr=7977 gr=[], dc=-31908 gc=[6, 6], db=1018120464 gb=[6, 3]
41: dr=8373 gr=[3], dc=-33492 gc=[30, 2], db=1121714064 gb=[30, 3]
42: dr=8745 gr=[6, 2], dc=-34980 gc=[6, 2, 2, 2], db=1223600400 gb=[12, 6, 2, 2]
43: dr=8828 gr=[3], dc=-2207 gc=[39], db=77933584 gb=[39, 3]
44: dr=8837 gr=[3], dc=-35348 gc=[78], db=1249481104 gb=[39, 3]
45: dr=8905 gr=[6, 2], dc=-35620 gc=[12, 2, 2], db=1268784400 gb=[12, 6, 2, 2]
46: dr=9069 gr=[], dc=-36276 gc=[12, 6], db=1315948176 gb=[12, 3]
47: dr=9149 gr=[3], dc=-36596 gc=[66, 2], db=1339267216 gb=[66, 3]
48: dr=9385 gr=[2], dc=-37540 gc=[12, 6], db=1409251600 gb=[12, 6]
49: dr=9413 gr=[3], dc=-37652 gc=[102], db=1417673104 gb=[51, 3]
50: dr=9497 gr=[], dc=-37988 gc=[24, 3], db=1443088144 gb=[12, 3]
51: dr=9745 gr=[6], dc=-38980 gc=[24, 2], db=1519440400 gb=[12, 12]
52: dr=10173 gr=[], dc=-40692 gc=[6, 6], db=1655838864 gb=[6, 3]
53: dr=10353 gr=[6, 2], dc=-41412 gc=[6, 2, 2, 2], db=1714953744 gb=[12, 6, 2, 2]
54: dr=10540 gr=[6, 2], dc=-2635 gc=[6, 2], db=111091600 gb=[6, 6, 2]
55: dr=10865 gr=[6, 2], dc=-43460 gc=[24, 2, 2], db=1888771600 gb=[24, 6, 2, 2]
56: dr=11001 gr=[2], dc=-44004 gc=[6, 6, 2], db=1936352016 gb=[6, 6, 2]
57: dr=11032 gr=[6], dc=-11032 gc=[12, 2], db=486820096 gb=[12, 6]
58: dr=11137 gr=[6], dc=-44548 gc=[12, 2, 2], db=1984524304 gb=[24, 6]
59: dr=11321 gr=[15], dc=-45284 gc=[132], db=2050640656 gb=[330, 3]
60: dr=11656 gr=[12], dc=-11656 gc=[12, 2], db=543449344 gb=[24, 12]
61: dr=11697 gr=[6], dc=-46788 gc=[12, 2, 2], db=2189116944 gb=[24, 6]
62: dr=11777 gr=[3], dc=-47108 gc=[120], db=2219163664 gb=[60, 3]
63: dr=12013 gr=[2], dc=-48052 gc=[6, 6], db=2308994704 gb=[6, 6]
64: dr=12309 gr=[6], dc=-49236 gc=[24, 2, 2], db=2424183696 gb=[48, 6]
65: dr=12453 gr=[2], dc=-49812 gc=[6, 6, 2], db=2481235344 gb=[6, 6, 2]
66: dr=12481 gr=[], dc=-49924 gc=[12, 6], db=2492405776 gb=[12, 3]
67: dr=12552 gr=[6], dc=-12552 gc=[12, 2], db=630210816 gb=[24, 6]
68: dr=12577 gr=[3], dc=-50308 gc=[48], db=2530894864 gb=[24, 3]
69: dr=12632 gr=[3], dc=-12632 gc=[42], db=638269696 gb=[42, 3]
70: dr=12685 gr=[6], dc=-50740 gc=[12, 2, 2], db=2574547600 gb=[24, 6]
71: dr=12837 gr=[2], dc=-51348 gc=[6, 6, 2], db=2636617104 gb=[12, 6]
72: dr=12849 gr=[3], dc=-51396 gc=[30, 2], db=2641548816 gb=[30, 3]
73: dr=13153 gr=[3], dc=-52612 gc=[24, 2], db=2768022544 gb=[24, 3]
74: dr=13273 gr=[6], dc=-53092 gc=[24, 2], db=2818760464 gb=[24, 6]
75: dr=13537 gr=[3], dc=-54148 gc=[60], db=2932005904 gb=[30, 3]
76: dr=13693 gr=[15], dc=-54772 gc=[42], db=2999971984 gb=[105, 3]
77: dr=13861 gr=[3], dc=-55444 gc=[30, 2], db=3074037136 gb=[30, 3]
78: dr=13897 gr=[6], dc=-55588 gc=[24, 2], db=3090025744 gb=[12, 12]
79: dr=14089 gr=[6], dc=-56356 gc=[48, 2], db=3175998736 gb=[24, 6, 2]
80: dr=14129 gr=[3], dc=-56516 gc=[84, 2], db=3194058256 gb=[84, 3]
81: dr=14165 gr=[6], dc=-56660 gc=[66, 2], db=3210355600 gb=[66, 6]
82: dr=14408 gr=[6], dc=-14408 gc=[48], db=830361856 gb=[48, 6]
83: dr=14609 gr=[3], dc=-58436 gc=[60, 2], db=3414766096 gb=[60, 3]
84: dr=15033 gr=[3], dc=-60132 gc=[30, 2], db=3615857424 gb=[30, 3]
85: dr=15049 gr=[2], dc=-60196 gc=[12, 6], db=3623558416 gb=[12, 6]
86: dr=15061 gr=[3], dc=-60244 gc=[66], db=3629339536 gb=[33, 3]
87: dr=15089 gr=[3], dc=-60356 gc=[84, 2], db=3642846736 gb=[84, 3]
88: dr=15212 gr=[3], dc=-3803 gc=[15], db=231404944 gb=[15, 3]
89: dr=15265 gr=[6], dc=-61060 gc=[24, 2, 2], db=3728323600 gb=[48, 6]
90: dr=15544 gr=[2], dc=-15544 gc=[6, 6], db=966463744 gb=[6, 6]
91: dr=15641 gr=[3], dc=-62564 gc=[192], db=3914254096 gb=[96, 3]
92: dr=15733 gr=[3], dc=-62932 gc=[42], db=3960436624 gb=[21, 3]
93: dr=15757 gr=[3], dc=-63028 gc=[24, 2], db=3972528784 gb=[24, 3]
94: dr=15773 gr=[3], dc=-63092 gc=[78], db=3980600464 gb=[39, 3]
95: dr=15820 gr=[6, 2], dc=-3955 gc=[6, 2], db=250272400 gb=[6, 6, 2]
96: dr=15848 gr=[6], dc=-15848 gc=[30, 2], db=1004636416 gb=[30, 6]
97: dr=15873 gr=[6, 2], dc=-63492 gc=[6, 2, 2, 2], db=4031234064 gb=[12, 6, 2, 2]
98: dr=16045 gr=[6], dc=-64180 gc=[24, 2], db=4119072400 gb=[24, 6]
99: dr=16049 gr=[], dc=-64196 gc=[30, 6], db=4121126416 gb=[30, 3]
100: dr=16108 gr=[], dc=-4027 gc=[3, 3], db=259467664 gb=[3, 3]
101: dr=16301 gr=[], dc=-65204 gc=[78, 3], db=4251561616 gb=[39, 3]
102: dr=16376 gr=[6], dc=-16376 gc=[48, 2], db=1072693504 gb=[48, 6, 2]
103: dr=16609 gr=[12], dc=-66436 gc=[24, 4], db=4413742096 gb=[24, 12, 2]
104: dr=16649 gr=[3], dc=-66596 gc=[204], db=4435027216 gb=[102, 3]
105: dr=16913 gr=[12], dc=-67652 gc=[60, 2], db=4576793104 gb=[120, 6]
106: dr=17081 gr=[6], dc=-68324 gc=[42, 2, 2], db=4668168976 gb=[84, 6]
107: dr=17132 gr=[3], dc=-4283 gc=[21], db=293505424 gb=[21, 3]
108: dr=17477 gr=[3], dc=-69908 gc=[114], db=4887128464 gb=[57, 3]
109: dr=17561 gr=[2], dc=-70244 gc=[12, 12], db=4934219536 gb=[12, 6, 2]
110: dr=17736 gr=[6], dc=-17736 gc=[24, 2], db=1258262784 gb=[48, 6]
111: dr=18044 gr=[6], dc=-4511 gc=[84], db=325585936 gb=[84, 3]
112: dr=18521 gr=[3], dc=-74084 gc=[228], db=5488439056 gb=[114, 3]
113: dr=18796 gr=[6], dc=-4699 gc=[12], db=353289616 gb=[12, 3]
114: dr=18853 gr=[6], dc=-75412 gc=[24, 2], db=5686969744 gb=[24, 6]
115: dr=19020 gr=[6, 2], dc=-4755 gc=[6, 2], db=361760400 gb=[6, 6, 2]
116: dr=19084 gr=[6], dc=-4771 gc=[12], db=364199056 gb=[12, 3]
117: dr=19112 gr=[6], dc=-19112 gc=[42], db=1461074176 gb=[42, 3]
118: dr=19544 gr=[6], dc=-19544 gc=[42, 2], db=1527871744 gb=[42, 6]
119: dr=19545 gr=[2], dc=-78180 gc=[6, 6, 2], db=6112112400 gb=[12, 6]
120: dr=19605 gr=[6], dc=-78420 gc=[24, 2, 2], db=6149696400 gb=[48, 6]
121: dr=19628 gr=[6], dc=-4907 gc=[12], db=385258384 gb=[12, 3]
122: dr=19677 gr=[2], dc=-78708 gc=[6, 6, 2], db=6194949264 gb=[6, 6, 2]
123: dr=19877 gr=[6], dc=-79508 gc=[30, 2, 2], db=6321522064 gb=[60, 6]
124: dr=19976 gr=[6], dc=-19976 gc=[48, 2], db=1596162304 gb=[96, 6]
125: dr=20044 gr=[3], dc=-5011 gc=[21], db=401761936 gb=[21, 3]
126: dr=20073 gr=[3], dc=-80292 gc=[30, 2], db=6446805264 gb=[30, 3]
127: dr=20093 gr=[3], dc=-80372 gc=[78, 2], db=6459658384 gb=[78, 3]
128: dr=20129 gr=[], dc=-80516 gc=[60, 3], db=6482826256 gb=[30, 3]
129: dr=20353 gr=[3], dc=-81412 gc=[48], db=6627913744 gb=[24, 3]
130: dr=20380 gr=[6], dc=-5095 gc=[48], db=415344400 gb=[48, 3]
131: dr=20396 gr=[3], dc=-5099 gc=[39], db=415996816 gb=[39, 3]
132: dr=20521 gr=[3], dc=-82084 gc=[84], db=6737783056 gb=[42, 3]
133: dr=20568 gr=[2], dc=-20568 gc=[6, 6], db=1692170496 gb=[6, 6, 2]
134: dr=20712 gr=[6], dc=-20712 gc=[24, 2], db=1715947776 gb=[24, 6]
135: dr=20829 gr=[6], dc=-83316 gc=[30, 2, 2], db=6941555856 gb=[60, 6]
136: dr=20865 gr=[6, 2], dc=-83460 gc=[12, 2, 2, 2], db=6965571600 gb=[24, 6, 2, 2]
137: dr=21224 gr=[2], dc=-21224 gc=[12, 6], db=1801832704 gb=[12, 6]
138: dr=21337 gr=[3], dc=-85348 gc=[30, 2], db=7284281104 gb=[30, 3]
139: dr=21449 gr=[2], dc=-85796 gc=[24, 6], db=7360953616 gb=[12, 6, 2]
140: dr=21469 gr=[3], dc=-85876 gc=[60, 2], db=7374687376 gb=[60, 3]
141: dr=21557 gr=[3], dc=-86228 gc=[114], db=7435267984 gb=[57, 3]
142: dr=21589 gr=[3], dc=-86356 gc=[102], db=7457358736 gb=[51, 3]
143: dr=21713 gr=[3], dc=-86852 gc=[168], db=7543269904 gb=[84, 3]
144: dr=21724 gr=[3], dc=-5431 gc=[57], db=471932176 gb=[57, 3]
145: dr=21737 gr=[3], dc=-86948 gc=[96], db=7559954704 gb=[48, 3]
146: dr=21809 gr=[6], dc=-87236 gc=[84, 2], db=7610119696 gb=[42, 6, 2]
147: dr=22229 gr=[3], dc=-88916 gc=[150], db=7906055056 gb=[75, 3]
148: dr=22341 gr=[6], dc=-89364 gc=[24, 2, 2], db=7985924496 gb=[48, 6]
149: dr=22377 gr=[3], dc=-89508 gc=[42, 2], db=8011682064 gb=[42, 3]
150: dr=22380 gr=[6, 2], dc=-5595 gc=[6, 2], db=500864400 gb=[6, 6, 2]
151: dr=22481 gr=[], dc=-89924 gc=[60, 3], db=8086325776 gb=[30, 3]
152: dr=22709 gr=[3], dc=-90836 gc=[114], db=8251178896 gb=[57, 3]
153: dr=22721 gr=[3], dc=-90884 gc=[264], db=8259901456 gb=[132, 3]
154: dr=22769 gr=[3], dc=-91076 gc=[204], db=8294837776 gb=[102, 3]
155: dr=22873 gr=[12], dc=-91492 gc=[12, 4], db=8370786064 gb=[12, 12, 2]
156: dr=22909 gr=[3], dc=-91636 gc=[66, 2], db=8397156496 gb=[66, 3]
157: dr=22965 gr=[2], dc=-91860 gc=[12, 6, 2], db=8438259600 gb=[24, 6]
158: dr=23033 gr=[3], dc=-92132 gc=[96, 2], db=8488305424 gb=[96, 3]
159: dr=23109 gr=[3], dc=-92436 gc=[48, 2], db=8544414096 gb=[48, 3]
160: dr=23165 gr=[4, 2], dc=-92660 gc=[12, 6, 2], db=8585875600 gb=[12, 12, 2, 2]
161: dr=23321 gr=[3], dc=-93284 gc=[168], db=8701904656 gb=[84, 3]
162: dr=23592 gr=[6], dc=-23592 gc=[24, 2], db=2226329856 gb=[24, 6]
163: dr=23605 gr=[2], dc=-94420 gc=[12, 6], db=8915136400 gb=[12, 6]
164: dr=23665 gr=[6], dc=-94660 gc=[48, 2], db=8960515600 gb=[48, 6]
165: dr=23689 gr=[3], dc=-94756 gc=[120], db=8978699536 gb=[60, 3]
166: dr=23752 gr=[6], dc=-23752 gc=[24], db=2256630016 gb=[24, 6]
167: dr=23816 gr=[6, 2], dc=-23816 gc=[42, 2], db=2268807424 gb=[42, 6, 2]
168: dr=23953 gr=[12], dc=-95812 gc=[24, 4], db=9179939344 gb=[12, 12, 4]
169: dr=23957 gr=[3], dc=-95828 gc=[138], db=9183005584 gb=[69, 3]
170: dr=24109 gr=[], dc=-96436 gc=[30, 3], db=9299902096 gb=[15, 3]
171: dr=24281 gr=[3], dc=-97124 gc=[228], db=9433071376 gb=[114, 3]
172: dr=24513 gr=[3], dc=-98052 gc=[42, 2], db=9614194704 gb=[42, 3]
173: dr=24577 gr=[3], dc=-98308 gc=[24, 2], db=9664462864 gb=[24, 3]
174: dr=24617 gr=[3], dc=-98468 gc=[96, 2], db=9695947024 gb=[96, 3]
175: dr=24904 gr=[2], dc=-24904 gc=[12, 6], db=2480836864 gb=[24, 6]
176: dr=24952 gr=[3], dc=-24952 gc=[24], db=2490409216 gb=[24, 3]
177: dr=25009 gr=[2], dc=-100036 gc=[24, 6], db=10007201296 gb=[12, 6, 2]
178: dr=25249 gr=[3], dc=-100996 gc=[48, 2], db=10200192016 gb=[48, 3]
179: dr=25464 gr=[6], dc=-25464 gc=[42, 2], db=2593661184 gb=[42, 6]
180: dr=25537 gr=[3], dc=-102148 gc=[48], db=10434213904 gb=[24, 3]
181: dr=25945 gr=[6], dc=-103780 gc=[48, 2], db=10770288400 gb=[24, 12]
182: dr=25949 gr=[6], dc=-103796 gc=[60, 2, 2], db=10773609616 gb=[60, 6, 2]
183: dr=26093 gr=[6], dc=-104372 gc=[48, 2], db=10893514384 gb=[48, 6]
184: dr=26232 gr=[6], dc=-26232 gc=[30, 2], db=2752471296 gb=[30, 6]
185: dr=26284 gr=[3], dc=-6571 gc=[15], db=690848656 gb=[15, 3]
186: dr=26305 gr=[8], dc=-105220 gc=[12, 6], db=11071248400 gb=[48, 6]
187: dr=26332 gr=[2], dc=-6583 gc=[12, 3], db=693374224 gb=[12, 3]
188: dr=26421 gr=[3], dc=-105684 gc=[60, 2], db=11169107856 gb=[60, 3]
189: dr=26489 gr=[3], dc=-105956 gc=[120], db=11226673936 gb=[60, 3]
190: dr=26492 gr=[6], dc=-6623 gc=[42], db=701826064 gb=[42, 3]
191: dr=26760 gr=[2, 2], dc=-26760 gc=[6, 6, 2], db=2864390400 gb=[12, 6, 2]
192: dr=26821 gr=[3], dc=-107284 gc=[78], db=11509856656 gb=[39, 3]
193: dr=26997 gr=[3], dc=-107988 gc=[48, 2], db=11661408144 gb=[48, 3]
194: dr=27004 gr=[6], dc=-6751 gc=[66], db=729216016 gb=[66, 3]
195: dr=27049 gr=[3], dc=-108196 gc=[42, 2], db=11706374416 gb=[42, 3]
196: dr=27065 gr=[6], dc=-108260 gc=[96, 2], db=11720227600 gb=[96, 6]
197: dr=27192 gr=[6, 2], dc=-27192 gc=[12, 2, 2], db=2957619456 gb=[12, 12, 2]
198: dr=27213 gr=[6], dc=-108852 gc=[24, 2, 2], db=11848757904 gb=[24, 6, 2]
199: dr=27336 gr=[6, 2], dc=-27336 gc=[12, 2, 2], db=2989027584 gb=[24, 6, 2, 2]
200: dr=27437 gr=[3], dc=-109748 gc=[114], db=12044623504 gb=[57, 3]
201: dr=27548 gr=[6], dc=-6887 gc=[78], db=758892304 gb=[78, 6]
202: dr=27617 gr=[3], dc=-110468 gc=[132], db=12203179024 gb=[66, 3]
203: dr=27640 gr=[2], dc=-27640 gc=[6, 6], db=3055878400 gb=[6, 6]
204: dr=27773 gr=[], dc=-111092 gc=[30, 3], db=12341432464 gb=[15, 3]
205: dr=28309 gr=[3], dc=-113236 gc=[114], db=12822391696 gb=[57, 3]
206: dr=28389 gr=[3], dc=-113556 gc=[66, 2], db=12894965136 gb=[66, 3]
207: dr=28428 gr=[6, 2], dc=-7107 gc=[6, 2], db=808151184 gb=[12, 6]
208: dr=28441 gr=[6], dc=-113764 gc=[24, 2, 2], db=12942247696 gb=[24, 6, 2]
209: dr=28945 gr=[2], dc=-115780 gc=[6, 6, 2], db=13405008400 gb=[12, 6]
210: dr=29281 gr=[6], dc=-117124 gc=[24, 2, 2], db=13718031376 gb=[24, 6, 2]
211: dr=29836 gr=[3], dc=-7459 gc=[15], db=890186896 gb=[15, 3]
The 90 bicyclic biquadratic fields B of type (9,3), 0 < D < 3*104
are given in the following table.
(Output format is Counter: d(R) Cl(R), d(C) Cl(C), d(B) Cl(B).)
1: dr=1373 gr=[3], dc=-5492 gc=[18], db=30162064 gb=[9, 3]
2: dr=2557 gr=[3], dc=-10228 gc=[18], db=104611984 gb=[9, 3]
3: dr=3889 gr=[3], dc=-15556 gc=[36], db=241989136 gb=[18, 3]
4: dr=4001 gr=[3], dc=-16004 gc=[72], db=256128016 gb=[36, 3]
5: dr=4765 gr=[6], dc=-19060 gc=[18, 2], db=363283600 gb=[18, 6]
6: dr=4857 gr=[3], dc=-19428 gc=[18, 2], db=377447184 gb=[18, 3]
7: dr=5297 gr=[3], dc=-21188 gc=[72], db=448931344 gb=[36, 3]
8: dr=5521 gr=[9], dc=-22084 gc=[60], db=487703056 gb=[90, 3]
9: dr=5621 gr=[6], dc=-22484 gc=[18, 2, 2], db=505530256 gb=[18, 6, 2]
10: dr=5853 gr=[3], dc=-23412 gc=[18, 2], db=548121744 gb=[18, 3]
11: dr=6396 gr=[6, 2], dc=-1599 gc=[18, 2], db=40908816 gb=[18, 6, 2]
12: dr=7057 gr=[21], dc=-28228 gc=[36], db=796819984 gb=[126, 3]
13: dr=7117 gr=[3], dc=-28468 gc=[18, 2], db=810427024 gb=[18, 3]
14: dr=7528 gr=[6], dc=-7528 gc=[18], db=226683136 gb=[18, 3]
15: dr=8277 gr=[6], dc=-33108 gc=[18, 2, 2], db=1096139664 gb=[18, 6, 2]
16: dr=9073 gr=[3], dc=-36292 gc=[18, 2], db=1317109264 gb=[18, 3]
17: dr=10941 gr=[6], dc=-43764 gc=[18, 2, 2], db=1915287696 gb=[18, 6, 2]
18: dr=10949 gr=[3], dc=-43796 gc=[126], db=1918089616 gb=[63, 3]
19: dr=11057 gr=[3], dc=-44228 gc=[72], db=1956115984 gb=[36, 3]
20: dr=11608 gr=[3], dc=-11608 gc=[18], db=538982656 gb=[18, 3]
21: dr=11641 gr=[9], dc=-46564 gc=[24, 2], db=2168206096 gb=[72, 3]
22: dr=12140 gr=[6], dc=-3035 gc=[18], db=147379600 gb=[18, 3]
23: dr=12197 gr=[3], dc=-48788 gc=[126], db=2380268944 gb=[63, 3]
24: dr=12409 gr=[9], dc=-49636 gc=[60], db=2463732496 gb=[90, 3]
25: dr=12652 gr=[3], dc=-3163 gc=[9], db=160073104 gb=[9, 3]
26: dr=12657 gr=[9], dc=-50628 gc=[30, 2], db=2563194384 gb=[90, 3]
27: dr=13069 gr=[9], dc=-52276 gc=[30, 2], db=2732780176 gb=[90, 3]
28: dr=13196 gr=[], dc=-3299 gc=[9, 3], db=174134416 gb=[9, 3]
29: dr=13549 gr=[12], dc=-54196 gc=[36, 2], db=2937206416 gb=[36, 12]
30: dr=13577 gr=[3], dc=-54308 gc=[72], db=2949358864 gb=[36, 3]
31: dr=14033 gr=[], dc=-56132 gc=[36, 3], db=3150801424 gb=[18, 3]
32: dr=14056 gr=[6], dc=-14056 gc=[18, 2], db=790284544 gb=[18, 6]
33: dr=14141 gr=[3], dc=-56564 gc=[72, 2], db=3199486096 gb=[72, 3]
34: dr=14680 gr=[6], dc=-14680 gc=[18, 2], db=862009600 gb=[18, 6]
35: dr=14969 gr=[3], dc=-59876 gc=[180], db=3585135376 gb=[90, 3]
36: dr=15689 gr=[6], dc=-62756 gc=[72, 2], db=3938315536 gb=[72, 6]
37: dr=16141 gr=[3], dc=-64564 gc=[90], db=4168510096 gb=[45, 3]
38: dr=16321 gr=[9], dc=-65284 gc=[42, 2], db=4262000656 gb=[126, 3]
39: dr=16645 gr=[12], dc=-66580 gc=[36, 2], db=4432896400 gb=[36, 12]
40: dr=16673 gr=[3], dc=-66692 gc=[144], db=4447822864 gb=[72, 3]
41: dr=16737 gr=[6], dc=-66948 gc=[18, 2, 2], db=4482034704 gb=[36, 6]
42: dr=17176 gr=[6], dc=-17176 gc=[18, 2], db=1180059904 gb=[18, 6, 2]
43: dr=17448 gr=[6], dc=-17448 gc=[18, 2], db=1217730816 gb=[18, 6]
44: dr=18492 gr=[6, 2], dc=-4623 gc=[18, 2], db=341954064 gb=[36, 6]
45: dr=19045 gr=[6, 2], dc=-76180 gc=[18, 2, 2], db=5803392400 gb=[18, 6, 2, 2]
46: dr=19261 gr=[6], dc=-77044 gc=[18, 2, 2], db=5935777936 gb=[18, 6, 2]
47: dr=19741 gr=[3], dc=-78964 gc=[36, 2], db=6235313296 gb=[36, 3]
48: dr=19869 gr=[6], dc=-79476 gc=[36, 2, 2], db=6316434576 gb=[72, 6]
49: dr=19885 gr=[6, 2], dc=-79540 gc=[18, 2, 2], db=6326611600 gb=[18, 6, 2, 2]
50: dr=19949 gr=[3], dc=-79796 gc=[198], db=6367401616 gb=[99, 3]
51: dr=20033 gr=[6], dc=-80132 gc=[36, 2, 2], db=6421137424 gb=[72, 6]
52: dr=20168 gr=[12], dc=-20168 gc=[36], db=1626992896 gb=[36, 6]
53: dr=20252 gr=[6], dc=-5063 gc=[36], db=410143504 gb=[36, 3]
54: dr=21101 gr=[3], dc=-84404 gc=[198], db=7124035216 gb=[99, 3]
55: dr=21109 gr=[6], dc=-84436 gc=[18, 2, 2], db=7129438096 gb=[36, 6]
56: dr=21293 gr=[3], dc=-85172 gc=[90, 2], db=7254269584 gb=[90, 3]
57: dr=22168 gr=[6], dc=-22168 gc=[18, 2], db=1965680896 gb=[18, 6, 2]
58: dr=22485 gr=[6], dc=-89940 gc=[18, 2, 2], db=8089203600 gb=[36, 6]
59: dr=22492 gr=[9], dc=-5623 gc=[33], db=505890064 gb=[99, 3]
60: dr=22812 gr=[2], dc=-5703 gc=[18, 3], db=520387344 gb=[18, 3]
61: dr=23249 gr=[3], dc=-92996 gc=[90, 2], db=8648256016 gb=[90, 3]
62: dr=23497 gr=[3], dc=-93988 gc=[72], db=8833744144 gb=[36, 3]
63: dr=24433 gr=[12], dc=-97732 gc=[36, 2], db=9551543824 gb=[72, 6]
64: dr=25708 gr=[3], dc=-6427 gc=[9], db=660901264 gb=[9, 3]
65: dr=26556 gr=[6], dc=-6639 gc=[90], db=705221136 gb=[90, 3]
66: dr=26744 gr=[3], dc=-26744 gc=[72], db=2860966144 gb=[72, 3]
67: dr=26761 gr=[3], dc=-107044 gc=[36, 2], db=11458417936 gb=[36, 3]
68: dr=26789 gr=[2], dc=-107156 gc=[18, 6, 2], db=11482408336 gb=[18, 6, 2]
69: dr=26853 gr=[3], dc=-107412 gc=[36, 2], db=11537337744 gb=[36, 3]
70: dr=27245 gr=[6], dc=-108980 gc=[72, 2], db=11876640400 gb=[72, 6]
71: dr=27329 gr=[3], dc=-109316 gc=[144], db=11949987856 gb=[72, 3]
72: dr=27340 gr=[6], dc=-6835 gc=[18], db=747475600 gb=[18, 3]
73: dr=27349 gr=[9], dc=-109396 gc=[48, 2], db=11967484816 gb=[144, 3]
74: dr=27409 gr=[3], dc=-109636 gc=[72], db=12020052496 gb=[36, 3]
75: dr=27633 gr=[6], dc=-110532 gc=[18, 2, 2], db=12217323024 gb=[36, 6]
76: dr=27649 gr=[], dc=-110596 gc=[18, 6], db=12231475216 gb=[18, 3]
77: dr=27656 gr=[2], dc=-27656 gc=[36, 3], db=3059417344 gb=[36, 6]
78: dr=27713 gr=[12], dc=-110852 gc=[36, 2, 2], db=12288165904 gb=[72, 12]
79: dr=28137 gr=[6], dc=-112548 gc=[18, 2, 2], db=12667052304 gb=[18, 6, 2]
80: dr=28477 gr=[], dc=-113908 gc=[18, 3], db=12975032464 gb=[9, 3]
81: dr=28504 gr=[6], dc=-28504 gc=[18, 2], db=3249912064 gb=[18, 6]
82: dr=28613 gr=[6], dc=-114452 gc=[36, 2, 2], db=13099260304 gb=[72, 6]
83: dr=28901 gr=[9], dc=-115604 gc=[222], db=13364284816 gb=[333, 3]
84: dr=28904 gr=[18], dc=-28904 gc=[102], db=3341764864 gb=[306, 3]
85: dr=29036 gr=[6, 2], dc=-7259 gc=[18, 2], db=843089296 gb=[18, 6, 2]
86: dr=29048 gr=[3], dc=-29048 gc=[72], db=3375145216 gb=[72, 3]
87: dr=29165 gr=[6], dc=-116660 gc=[36, 2, 2], db=13609555600 gb=[72, 6]
88: dr=29397 gr=[6], dc=-117588 gc=[18, 2, 2], db=13826937744 gb=[18, 6, 2]
89: dr=29469 gr=[6, 2], dc=-117876 gc=[18, 2, 2, 2], db=13894751376 gb=[36, 12, 2]
90: dr=29957 gr=[2], dc=-119828 gc=[18, 6], db=14358749584 gb=[18, 6]

3.2. Triadic Quantum Class Group G32(B) of B = Q(D1/2,(-1)1/2),
the bicyclic biquadratic field of Gauss-Dirichlet-Hilbert type

With the aid of MAGMA V2.18-9 , we have computed the
triadic quantum class group G = G32(B)
with abelianization of diamond type (3,3), resp. of double layered type (9,3),
for the 211, resp. 90, bicyclic biquadratic fields B = Q(D1/2,(-1)1/2)
with 3-class group Cl3(B) of type (3,3), resp. (9,3), and 0 < D < 3*104,
in March and August 2012.
The discriminants d(B) of these fields B are contained in the range
3579664 ≤ d(B) ≤ 14358749584.

Our algorithm is based upon the triadic TTT (transfer target type)
τ = (str(Cl3(N1)),…,str(Cl3(N4))).

3.2.1. Intrinsic (Genuine) Triadic Quantum Class Groups

Roughly 80% of the 211 bicyclic biquadratic fields B = Q(D1/2,(-1)1/2)
with 3-class group Cl3(B) of type (3,3) and 0 < D < 3*104
are composita of dual quadratic fields R and C of equal 3-class rank 1.
Their triadic quantum class groups G = G32(B) are intrinsic, genuine
invariants of the bicyclic biquadratic fields B.
No. Discriminant 3-Class Group of Cohomology Transfer Kernel Quantum 3-Class
D = d(R) B L1 L2 L3 L4 N1 N2 N3 N4 F31(B) ε1 Type Type (TKT) Group, G32(B)
Coclass 1 (GS)
1 473 (3,3) 3 3 3 3 (9,3) (3,3) (3,3) (3,3) (3,3) 0 (δααα) a.3 (2000) < 81,8 >
5 1937 (3,3) 3 3 3 3 (3,3,3) (3,3) (3,3) (3,3) (3,3) 1 (δααα) a.3* (2000) < 81,7 >
Coclass 1 (ES 1)
10 2993 (3,3) 9 3 3 3 (27,9) (3,3) (3,3) (3,3) (9,9) 0 (δααα) a.3 ↑ (2000) < 729,97|98 >
Coclass 2 (GS)
8 2713 (3,3) 3 3 3 3 (3,3,3) (3,3,3) (9,3) (3,3,3) (9,3,3) 3 (δδδδ) H.4 (4443) < 729,45 > = K
11 3305 (3,3) 9 3 3 3 (9,9) (9,3) (9,3) (9,3) (9,3,3) 0 (αδδδ) c.21 (0231) < 729,54 > = U
16 3941 (3,3) 3 3 3 3 (9,3) (9,3) (9,3) (9,3) (3,3,3,3) 0 (δδδδ) G.19 (2143) < 729,57 > = W
73 13153 (3,3) 9 3 3 3 (9,9) (9,3) (3,3,3) (9,3) (9,3,3) 1 (αδδδ) c.18 (0313) < 729,49 > = Q
Coclass 2 (ES 1)
38 7665 (3,3) 9 3 3 3 (27,9) (9,3) (9,3) (9,3) (27,9,3) 0 (δδδδ) G.16 ↑ (4231) < 6561,# >
89 15265 (3,3) 9 3 3 3 (27,9) (9,3) (3,3,3) (9,3) (27,9,3) 1 (δδδδ) H.4 ↑ (3313) < 6561,# >
Coclass 4 (2*ES 1)
28 5912 (3,3) 9 9 3 3 (27,9) (27,9) (3,3,3) (3,3,3) (27,9,9,3) 2 (δδδδ) G.16 ↑↑ (1243) < 59049,# >
70 12685 (3,3) 9 9 3 3 (27,9) (27,9) (3,3,3) (3,3,3) (27,9,9,3) 2 (δδδδ) G.19 ↑↑ (2143) < 59049,# >

3.2.2. Inherited (Lifted) Triadic Quantum Class Groups

About 20% of the 211 bicyclic biquadratic fields B = Q(D1/2,(-1)1/2)
with 3-class group Cl3(B) of type (3,3) and 0 < D < 3*104
are composita of a real quadratic field R of 3-rank 0 and its dual complex quadratic field C of 3-rank 2.
In this case, the triadic quantum class group G = G32(B) is inherited
from the quadratic subfield C by lifting the entire 3-class field tower isomorphically from C to B.
No. Discriminant 3-Class Group of Cohomology Transfer Kernel Quantum 3-Class
D = d(R) B L1 L2 L3 L4 N1 N2 N3 N4 F31(B) ε1 Type Type (TKT) Group, G32(B)
Coclass 2 (GS)
15 3896 (3,3) 3 3 3 3 (3,3,3) (3,3,3) (9,3) (3,3,3) (9,3,3) 3 (δδδδ) H.4 (4443) < 729,45 > = K
21 5069 (3,3) 3 3 3 3 (3,3,3) (9,3) (3,3,3) (9,3) (3,3,3) 2 (δδδδ) D.5 (4224) < 243,7 >
52 10173 (3,3) 3 3 3 3 (3,3,3) (9,3) (9,3) (9,3) (3,3,3) 1 (δδδδ) D.10 (2241) < 243,5 >
66 12481 (3,3) 3 3 3 3 (9,3) (9,3) (9,3) (9,3) (3,3,3,3) 0 (δδδδ) G.19 (2143) < 729,57 > = W
Coclass 2 (ES 1)
7 2437 (3,3) 9 3 3 3 (27,9) (9,3) (9,3) (9,3) (9,9,3) 0 (δδδδ) E.9 ↑ (2231) < 2187,# >
24 5417 (3,3) 9 3 3 3 (27,9) (9,3) (3,3,3) (9,3) (27,9,3) 1 (δδδδ) H.4 ↑ (3313) < 6561,# >
32 6221 (3,3) 9 3 3 3 (27,9) (9,3) (9,3) (9,3) (27,9,3) 0 (δδδδ) G.16 ↑ (4231) < 6561,# >
71 12837 (3,3) 9 3 3 3 (27,9) (9,3) (3,3,3) (9,3) (9,9,3) 1 (δδδδ) E.14 ↑ (2313) < 2187,# >
90 15544 (3,3) 9 3 3 3 (27,9) (9,3) (3,3,3) (9,3) (9,9,3) 1 (δδδδ) E.6 ↑ (1313) < 2187,# >
Coclass 4 (2*ES)
37 6789 (3,3) 9 9 3 3 (27,9) (27,9) (3,3,3) (3,3,3) (9,9,9,3) 2 (δδδδ) F.11 ↑↑ (1143) < 19683,# >
40 7977 (3,3) 9 9 3 3 (27,9) (27,9) (3,3,3) (3,3,3) (9,9,9,3) 2 (δδδδ) F.12 ↑↑ (1343) < 19683,# >
Statistical evaluation of the first 100 bicyclic biquadratic fields of type (3,3):

A. Intrinsic (Genuine) Scenario (78 Fields):


1. Triadic Quantum Class Groups of Coclass 1:

1.0. Ground States (GS).
Two 3-groups of the stem of isoclinism family Φ3 are dominating:
There occur 25 cases (32%) of the GS of TKT a.3 (2000), starting with no. 1: d(R) = 473.
There occur 22 cases (28%) of TKT a.3* (2000), starting with no. 5: d(R) = 1937.

1.1. First Excited State (ES 1).
There occur 5 cases (6%) of ES 1 of TKT a.3↑ (2000), starting with no. 10: d(R) = 2993.

2. Triadic Quantum Class Groups of Coclass 2:

2.0. Ground States (GS).
There occur 7 cases (9%) of the GS of TKT H.4 (4443), starting with no. 8: d(R) = 2713.
There occur 5 cases (9%) of the GS of TKT G.19 (2143), starting with no. 16: d(R) = 3941.
There occur 3 cases (4%) of the GS of TKT c.21 (0231), starting with no. 11: d(R) = 3305.
There occur 4 cases (5%) of the GS of TKT c.18 (0313), starting with no. 43: d(R) = 8828.

2.1. First Excited State (ES 1).
There occur 4 cases (5%) of ES 1 of TKT G.16↑ (4231), starting with no. 38: d(R) = 7665.
There s a single case of ES 1 of TKT H.4↑ (3313), for no. 89: d(R) = 15265.

4. Triadic Quantum Class Groups of Coclass 4:

4.1. Double Excited State (2*ES 1).
There occurs a single case of (ES 1, ES 1) of TKT G.16↑↑ (1243), for no. 28: d(R) = 5912.
There occurs a single case of (ES 1, ES 1) of TKT G.19↑↑ (2143), for no. 70: d(R) = 12685.

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