7.1. Complex Base ObjectsAccording to the following table, computed at the University of Manitoba, Winnipeg,the existence of 94 heptadic quantum class groups G72(K) with abelianizations of diamond type (7,7) over complex base objects K = Q(D1/2) with 7-irregular discriminants D has been discovered down to the lower bound -106 < D < 0. (Output format is Counter.UnsignedDiscriminant:ClassNumber, comma separated.) |
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7.2. Real Base ObjectsRecently we started a search for analogous positive discriminants and discoveredthe existence of 17 heptadic quantum class groups G72(K) with abelianizations of diamond type (7,7) over real base objects K = Q(D1/2) with 7-irregular discriminants D extending to the upper bound 107 > D > 0. They are given in the following table. (Output format is Counter:Discriminant, separated by semicolons.) |
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7.3. Recall of Triadic TTTThe triadic TTT (transfer target type) τ = (str(Cl3(N1)),…,str(Cl3(N4))) is well known by our theory of nearly homocyclic and exotic 3-class groups. The first component τ(1) of the TTT is not able to distinguish between the transfer kernel types (TKT) a.2 (1000) and a.3 (2000), neither in the GS (ground state) nor in the ES (excited states). The following table shows the minimal representatives of all known GS and ES of TKTs for vertices of coclass graph G(3,1) among the 2576 triadic quantum class groups G32(K) with abelianizations of diamond type (3,3) over real base objects K = Q(D1/2) with 3-irregular discriminants 0 < D < 107, computed from January to March 2010 with the aid of PARI/GP V2.3.4 . The invariant ε1 denotes the cardinality #{ 1 ≤ i ≤ 4 | Cl3(Ni) ≅ (3,3,3) }. |
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7.4. Introduction of Heptadic TTTThe heptadic TTT (transfer target type) τ = (str(Cl7(N1)),…,str(Cl7(N8))) admits the separation of the GS (ground state) of TKT (transfer kernel type) a.2 (10000000), where τ(1) = str(Cl7(N1)) = (7,7,7) and a.3 (20000000), where τ(1) = str(Cl7(N1)) = (49,7). and of the first ES (excited state) of TKT a.2 ↑ (10000000), where τ(1) = str(Cl7(N1)) = (7,7,7,7,7) and a.3 ↑ (20000000), where τ(1) = str(Cl7(N1)) = (49,7,7,7). The reason is that, according to Theorem 2, for any Ni with 1 ≤ i ≤ 8, an elementary abelian structure (7,…,7) of 7-rank at most 6 implies a coarse TKT of Taussky type (A) and an exceptional structure (49,7,…,7) of 7-rank at most 5 implies a coarse TKT of Taussky type (B). However, the first component τ(1) of the TTT is not able to distinguish the second and higher ES (excited states) of these TKTs. 7.4.1. TTTs for Quantum Groups of Coclass 1The following table shows the minimal representatives of all GS of TKTs for vertices of coclass graph G(7,1) among the 17 heptadic quantum class groups G72(K) with abelianizations of diamond type (7,7) over real base objects K = Q(D1/2) with 7-irregular discriminants 0 < D < 107, computed in January 2012 with the aid of MAGMA V2.18-3 and valuable advice by C. Fieker . The invariant ε1 denotes the cardinality #{ 1 ≤ i ≤ 8 | Cl7(Ni) ≅ (7,7,7,7,7,7,7) }. Additionally, we need other invariants ε2 = #{ 1 ≤ i ≤ 8 | Cl7(Ni) ≅ (7,7,7) } and ε3 = #{ 1 ≤ i ≤ 8 | Cl7(Ni) ≅ (7,7,7,7,7) }. |
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Statistical evaluation: Heptadic Quantum Class Groups of Coclass 1: There occur 3 cases of the GS of TKT a.2 (10000000) for 3: D = 2713121, 9: 6872024, and 16: 9659661. The remaining 13 cases (76%) of the GS of TKT a.3 (20000000), starting with 1: 1633285, are clearly dominating. The 7-class field tower for these 16 cases has exact length 2. Heptadic Quantum Class Group of Coclass 2: There occurs a single case of the GS of coarse TKT (BBBBBBBB) for 10: 6986985. 7.4.2. TTTs for Quantum Groups of Coclass 2The possibilities for coclass 2 are more extensive than those for coclass 1. The following table shows the minimal representatives of all known GS and ES of TKTs for vertices on the coclass graph G(7,2) among the 94 heptadic quantum class groups G72(K) with abelianizations of diamond type (7,7) over complex base objects K = Q(D1/2) with 7-irregular discriminants -106 < D < 0, 32 cases with -5*105 < D < 0 computed in January 2012 with the aid of MAGMA V2.18-3 and valuable advice by C. Fieker , the extension to all 94 cases computed in November 2012 with the aid of MAGMA V2.18-12 . The invariant ε1 denotes the cardinality #{ 1 ≤ i ≤ 8 | Cl7(Ni) ≅ (7,7,7,7,7,7,7) }. Additionally, we need other invariants ε2 = #{ 1 ≤ i ≤ 8 | Cl7(Ni) ≅ (7,7,7) } and ε3 = #{ 1 ≤ i ≤ 8 | Cl7(Ni) ≅ (7,7,7,7,7) }. For first, resp. second, excited states (ES 1, resp. ES 2) of coclass 2, one of the 7-class groups Cl7(Li) of the non-Galois absolute septic subfields Li of Ni (unramified over K) is of 7-rank 2, resp. 7-rank 3, which shows impressively that the rank equation for p = 3, r3(Li) = r3(K)-1, by G. Gras and F. Gerth III, conjectured by T. Callahan in 1974, generalizes to a double inequality for p ≥ 5, rp(K)-1 ≤ rp(Li) ≤ (rp(K)-1)*(p-1)/2, as predicted, and partially proved, by R. Bölling and F. Lemmermeyer . |
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Statistical evaluation: Heptadic Quantum Class Groups of Coclass 2: 0. Ground States (GS). Ground states appear exclusively with sporadic, and mostly terminal, top vertices of G(7,2). There occur 9 cases (9.6%) of coarse TKT (ABBBBBBB), starting with 12: D = -227860, and 29 cases (31%) of cTKT (AABBBBBB), starting with 7: -183619. The remaining 40 cases (43%) of cTKT (BBBBBBBB), starting with 1: -63499 are clearly dominating. The 7-class field tower for the 9 cases of coarse TKT (ABBBBBBB) has exact length 2, since the metabelian 7-groups <16807,8> and <16807,9> are Schur σ-groups. 1. First Excited States (ES 1). There occur 2 cases of ES 1 of cTKT ↑ (BBBBBBBB) starting with 19: -272179; 3 cases of ES 1 of cTKT ↑ (AABBBBBB) with 7-class group of type (7,7,7,7,7) starting with 5: -159592; and 10 cases (11%) of ES 1 of cTKT ↑ (BABBBBBB), starting with 11: -227387 and clearly dominating. 2. Second Excited State (ES 2). There occurs a single case of ES 2 of cTKT ↑2 (?ABBBBBB), for 59: -673611. |
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