A247696 -id:A247696 - cp's OEIS frontend

A247692 Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type E.6 (1122), and second 3-class group G of odd nilpotency class cl(G)=2(n+2)+1.

15544, 268040, 1062708, 27629107

Offset: 0

Daniel Constantin Mayer, Sep 28 2014

The 3-principalization type (transfer kernel type, TKT) E.6 (1122) is not a permutation and has a single fixed point.
The nilpotency condition cl(G)=2n+5 for the second 3-class group is equivalent to a transfer target type, TTT (called IPAD by Boston, Bush and Hajir) of the shape [(3^{n+2},3^{n+3}),(3,3,3),(3,9)^2].
The second 3-class group G is a vertex of depth 1 on the coclass tree with root SmallGroup(243,6) contained in the coclass graph G(3,2).
All these fields possess a Hilbert 3-class field tower of exact length 3.
A247692 is an extremely sparse subsequence of A242878 and it is exceedingly hard to compute a(n) for n>0.

			For a(0)=15544, we have the ground state of TKT E.6 with TTT [(9,27),(3,3,3),(3,9)^2] and cl(G)=5.
For a(1)=268040, we have the first excited state of TKT E.6 with TTT [(27,81),(3,3,3),(3,9)^2] and cl(G)=7.
a(0) and a(1) are due to D. C. Mayer (2012).
a(2) and a(3) are due to N. Boston, M. R. Bush and F. Hajir (2013).

N. Boston, M. R. Bush, F. Hajir, Heuristics for p-class towers of imaginary quadratic fields, Math. Ann. (2013), Preprint: arXiv:1111.4679v1 [math.NT], 2011.
M. R. Bush and D. C. Mayer, 3-class field towers of exact length 3, J. Number Theory (2014), Preprint: arXiv:1312.0251v1 [math.NT], 2013.
D. C. Mayer, The second p-class group of a number field, arXiv:1403.3899 [math.NT], 2014; Int. J. Number Theory 8 (2012), no. 2, 471-505.
D. C. Mayer, Transfers of metabelian p-groups, arXiv:1403.3896 [math.GR], 2014; Monatsh. Math. 166 (3-4) (2012), 467-495.
D. C. Mayer, The distribution of second p-class groups on coclass graphs, arXiv:1403.3833 [math.NT], 2014; J. Théor. Nombres Bordeaux 25 (2) (2013), 401-456.
D. C. Mayer, Principalization algorithm via class group structure, J. Théor. Nombres Bordeaux (2014), Preprint: arXiv:1403.3839v1 [math.NT], 2014.
Daniel C. Mayer, Periodic sequences of p-class tower groups, arXiv:1504.00851, 2015.
Wikipedia, Artin transfer (group theory), Table 2

Cf. A242862, A242863, A242878 (supersequences), A247693, A247694, A247695, A247696, A247697 (disjoint sequences).

A247693 Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type E.14 (3122), and second 3-class group G of odd nilpotency class cl(G)=2(n+2)+1.

Original entry on oeis.org

16627, 262744, 4776071, 40059363

Offset: 0

Daniel Constantin Mayer, Sep 28 2014

The 3-principalization type (transfer kernel type, TKT) E.14 (3122) is not a permutation, contains a 3-cycle, and has no fixed points.
The nilpotency condition cl(G)=2n+5 for the second 3-class group is equivalent to a transfer target type, TTT (called IPAD by Boston, Bush and Hajir) of the shape [(3^{n+2},3^{n+3}),(3,3,3),(3,9)^2].
The second 3-class group G is one of two vertices of depth 1 on the coclass tree with root SmallGroup(243,6) contained in the coclass graph G(3,2).
All these fields possess a Hilbert 3-class field tower of exact length 3.
A247693 is an extremely sparse subsequence of A242878 and it is exceedingly hard to compute a(n) for n>0.

			For a(0)=16627, we have the ground state of TKT E.14 with TTT [(9,27),(3,3,3),(3,9)^2] and cl(G)=5.
For a(1)=262744, we have the first excited state of TKT E.14 with TTT [(27,81),(3,3,3),(3,9)^2] and cl(G)=7.
a(0) and a(1) are due to D. C. Mayer (2012).
a(2) and a(3) are due to N. Boston, M. R. Bush and F. Hajir (2013).

N. Boston, M. R. Bush, and F. Hajir, Heuristics for p-class towers of imaginary quadratic fields, Math. Ann. (2013), Preprint: arXiv:1111.4679v1 [math.NT], 2011.
M. R. Bush and D. C. Mayer, 3-class field towers of exact length 3, J. Number Theory (2014), Preprint: arXiv:1312.0251v1 [math.NT], 2013.
D. C. Mayer, The second p-class group of a number field, arXiv:1403.3899 [math.NT], 2014; Int. J. Number Theory 8 (2012), no. 2, 471-505.
D. C. Mayer, Transfers of metabelian p-groups, arXiv:1403.3896 [math.GR], 2014; Monatsh. Math. 166 (3-4) (2012), 467-495.
D. C. Mayer, The distribution of second p-class groups on coclass graphs, arXiv:1403.3833 [math.NT], 2014; J. Théor. Nombres Bordeaux 25 (2) (2013), 401-456.
D. C. Mayer, Principalization algorithm via class group structure, J. Théor. Nombres Bordeaux (2014), Preprint: arXiv:1403.3839v1 [math.NT], 2014.
D. C. Mayer, Periodic sequences of p-class tower groups, arXiv:1504.00851 [math.NT], 2015.
Wikipedia, Artin transfer (group theory), Table 2

Cf. A242862, A242863, A242878 (supersequences), A247692, A247694, A247695, A247696, A247697 (disjoint sequences).

A247694 Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type H.4 (2122), second 3-class group G of even nilpotency class cl(G)=2(n+3), and 3-class tower of unknown length at least 3.

Original entry on oeis.org

21668, 446788, 3843907, 52505588

Offset: 0

Daniel Constantin Mayer, Oct 12 2014

The 3-principalization type (transfer kernel type, TKT) H.4 (2122) is not a permutation, contains a transposition, and has no fixed point.
The nilpotency condition cl(G)=2n+6 for the second 3-class group is equivalent to a transfer target type, TTT (called IPAD by Boston, Bush and Hajir) of the shape [(3^{n+2},3^{n+3}),(3,3,3),(3,9)^2].
The second 3-class group G is one of two vertices of depth 2 on the coclass tree with root SmallGroup(243,6) contained in the coclass graph G(3,2).
The length of the Hilbert 3-class field tower of all these fields is completely unknown. Therefore, these discriminants are among the foremost challenges of future research, similarly as those of A242873, A247688, A247697.
A247694 is an extremely sparse subsequence of A242878 and it is exceedingly hard to compute a(n) for n>0.
The initial term a(0)=21668 has been recognized as a realization of TKT H.4 in the Dissertation of J. R. Brink(1984). However, Brink did not know that the TKT H.4 can also occur with second 3-class group G=SmallGroup(729,45) of nilpotency class cl(G)=4 and TTT [(3,3,3)^3,(3,9)]. Actually, D. C. Mayer (1991) was the first who proved that the integer 21668 is the smallest term of A247694 and does not belong to A242873.

			For a(0)=21668, we have the ground state of TKT H.4 with TTT [(9,27),(3,3,3),(3,9)^2] and cl(G)=6.
For a(1)=446788, we have the first excited state of TKT H.4 with TTT [(27,81),(3,3,3),(3,9)^2] and cl(G)=8.
a(0) and a(1) are due to D. C. Mayer (2012).
a(2) and a(3) are due to N. Boston, M. R. Bush and F. Hajir (2013).

J. R. Brink, The class field tower for imaginary quadratic number fields of type (3,3), Dissertation, The Ohio State University, 1984.
D. C. Mayer, Principalization in complex S_3 fields, Congressus Numerantium 80 (1991), 73-87. (Proceedings of the Twentieth Manitoba Conference on Numerical Mathematics and Computing, The University of Manitoba, Winnipeg, Manitoba, Canada, 1990.)

N. Boston, M. R. Bush and F. Hajir, Heuristics for p-class towers of imaginary quadratic fields, Preprint: arXiv:1111.4679v1 [math.NT], 2011, Math. Ann. (2013).
M. R. Bush and D. C. Mayer, 3-class field towers of exact length 3, Preprint: arXiv:1312.0251v1 [math.NT], 2013.
D. C. Mayer, The second p-class group of a number field, Int. J. Number Theory 8 (2) (2012), 471-505.
D. C. Mayer, The second p-class group of a number field
D. C. Mayer, Transfers of metabelian p-groups, Monatsh. Math. 166 (3-4) (2012), 467-495.
D. C. Mayer, Transfers of metabelian p-groups, arXiv:1403.3896 [math.GR], 2014.
D. C. Mayer, The distribution of second p-class groups on coclass graphs, J. Théor. Nombres Bordeaux 25 (2) (2013), 401-456.
D. C. Mayer, The distribution of second p-class groups on coclass graphs, arXiv:1403.3833 [math.NT], 2014.
D. C. Mayer, Principalization algorithm via class group structure, Preprint: arXiv:1403.3839v1 [math.NT], 2014.
Daniel C. Mayer, Periodic sequences of p-class tower groups, arXiv:1504.00851 [math.NT], 2015.
Wikipedia, Artin transfer (group theory), Table 2

Cf. A242862, A242863, A242878 (supersequences), A247692, A247693, A247695, A247696, A247697 (disjoint sequences).

A247695 Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type E.8 (2234), and second 3-class group G of odd nilpotency class cl(G)=2(n+2)+1.

Original entry on oeis.org

34867, 370740, 4087295, 19027947

Offset: 0

Daniel Constantin Mayer, Sep 28 2014

The 3-principalization type (transfer kernel type, TKT) E.8 (2234) is not a permutation and has three fixed points.
The nilpotency condition cl(G)=2n+5 for the second 3-class group is equivalent to a transfer target type, TTT (called IPAD by Boston, Bush and Hajir) of the shape [(3,9),(3^{n+2},3^{n+3}),(3,9)^2].
The second 3-class group G is a vertex of depth 1 on the coclass tree with root SmallGroup(243,8) contained in the coclass graph G(3,2).
All these fields possess a Hilbert 3-class field tower of exact length 3.
A247695 is an extremely sparse subsequence of A242878 and it is exceedingly hard to compute a(n) for n>0.

			For a(0)=34867, we have the ground state of TKT E.8 with TTT [(3,9),(9,27),(3,9)^2] and cl(G)=5.
For a(1)=370740, we have the first excited state of TKT E.8 with TTT [(3,9),(27,81),(3,9)^2] and cl(G)=7.
a(0) and a(1) are due to D. C. Mayer (2012).
a(2) and a(3) are due to N. Boston, M. R. Bush and F. Hajir (2013).

N. Boston, M. R. Bush and F. Hajir, Heuristics for p-class towers of imaginary quadratic fields, Preprint: arXiv:1111.4679v1 [math.NT], 2011, Math. Ann. (2013).
M. R. Bush and D. C. Mayer, 3-class field towers of exact length 3, Preprint: arXiv:1312.0251v1 [math.NT], 2013.
D. C. Mayer, The second p-class group of a number field, Int. J. Number Theory 8 (2) (2012), 471-505.
D. C. Mayer, The second p-class group of a number field, arXiv:1403.3899 [math.NT], 2014.
D. C. Mayer, Transfers of metabelian p-groups, Monatsh. Math. 166 (3-4) (2012), 467-495.
D. C. Mayer, Transfers of metabelian p-groups, arXiv:1403.3896 [math.GR], 2014.
D. C. Mayer, The distribution of second p-class groups on coclass graphs, J. Théor. Nombres Bordeaux 25 (2) (2013), 401-456.
D. C. Mayer, The distribution of second p-class groups on coclass graphs, arXiv:1403.3833 [math.NT], 2014.
D. C. Mayer, Principalization algorithm via class group structure, Preprint: arXiv:1403.3839v1 [math.NT], 2014.
Daniel C. Mayer, Periodic sequences of p-class tower groups, arXiv:1504.00851 [math.NT], 2015.
Wikipedia, Artin transfer (group theory), Table 2

Cf. A242862, A242863, A242878 (supersequences), A247692, A247693, A247694, A247696, A247697 (disjoint sequences).

A247697 Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type G.16 (2134), second 3-class group G of even nilpotency class cl(G)=2(n+3), and 3-class tower of unknown length at least 3.

Original entry on oeis.org

17131, 819743, 2244399, 30224744

Offset: 0

Daniel Constantin Mayer, Oct 12 2014

The 3-principalization type (transfer kernel type, TKT) G.16 (2134) is a permutation, contains a transposition, and has two fixed points.
The nilpotency condition cl(G)=2n+6 for the second 3-class group is equivalent to a transfer target type, TTT (called IPAD by Boston, Bush and Hajir) of the shape [(3,9),(3^{n+2},3^{n+3}),(3,9)^2].
The second 3-class group G is one of two vertices of depth 2 on the coclass tree with root SmallGroup(243,8) contained in the coclass graph G(3,2).
The length of the Hilbert 3-class field tower of all these fields is completely unknown. Therefore, these discriminants are among the foremost challenges of future research, similarly as those of A242873, A247688, A247694.
A247697 is an extremely sparse subsequence of A242878 and it is exceedingly hard to compute a(n) for n>0.

			For a(0)=17131, we have the ground state of TKT G.16 with TTT [(3,9),(9,27),(3,9)^2] and cl(G)=6.
For a(1)=819743, we have the first excited state of TKT G.16 with TTT [(3,9),(27,81),(3,9)^2] and cl(G)=8.
a(0) and a(1) are due to D. C. Mayer (2012).
a(2) and a(3) are due to N. Boston, M. R. Bush and F. Hajir (2013).

N. Boston, M. R. Bush and F. Hajir, Heuristics for p-class towers of imaginary quadratic fields, Preprint: arXiv:1111.4679v1 [math.NT], 2011; Math. Ann. (2013).
M. R. Bush and D. C. Mayer, 3-class field towers of exact length 3, Preprint: arXiv:1312.0251v1 [math.NT], 2013.
D. C. Mayer, The second p-class group of a number field, Int. J. Number Theory 8 (2) (2012), 471-505.
D. C. Mayer, The second p-class group of a number field, arXiv:1403.3899 [math.NT], 2014.
D. C. Mayer, Transfers of metabelian p-groups, Monatsh. Math. 166 (3-4) (2012), 467-495.
D. C. Mayer, Transfers of metabelian p-groups, arXiv:1403.3896 [math.GR], 2014.
D. C. Mayer, The distribution of second p-class groups on coclass graphs, J. Théor. Nombres Bordeaux 25 (2) (2013), 401-456.
D. C. Mayer, The distribution of second p-class groups on coclass graphs, arXiv:1403.3833 [math.NT], 2014.
D. C. Mayer, Principalization algorithm via class group structure, Preprint: arXiv:1403.3839v1 [math.NT], 2014.
Daniel C. Mayer, Periodic sequences of p-class tower groups, arXiv:1504.00851 [math.NT], 2015.
Wikipedia, Artin transfer (group theory), Table 2

Cf. A242862, A242863, A242878 (supersequences), A247692, A247693, A247694, A247695, A247696 (disjoint sequences).

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A247692 Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type E.6 (1122), and second 3-class group G of odd nilpotency class cl(G)=2(n+2)+1.

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A247693 Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type E.14 (3122), and second 3-class group G of odd nilpotency class cl(G)=2(n+2)+1.

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A247694 Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type H.4 (2122), second 3-class group G of even nilpotency class cl(G)=2(n+3), and 3-class tower of unknown length at least 3.

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A247695 Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type E.8 (2234), and second 3-class group G of odd nilpotency class cl(G)=2(n+2)+1.

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A247697 Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type G.16 (2134), second 3-class group G of even nilpotency class cl(G)=2(n+3), and 3-class tower of unknown length at least 3.

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