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%I #23 Aug 06 2020 04:24:01
%S 21668,446788,3843907,52505588
%N 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.
%C The 3-principalization type (transfer kernel type, TKT) H.4 (2122) is not a permutation, contains a transposition, and has no fixed point.
%C 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].
%C 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).
%C 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.
%C A247694 is an extremely sparse subsequence of A242878 and it is exceedingly hard to compute a(n) for n>0.
%C 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.
%D J. R. Brink, The class field tower for imaginary quadratic number fields of type (3,3), Dissertation, The Ohio State University, 1984.
%D 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.)
%H N. Boston, M. R. Bush and F. Hajir, <a href="http://arxiv.org/abs/1111.4679">Heuristics for p-class towers of imaginary quadratic fields</a>, Preprint: arXiv:1111.4679v1 [math.NT], 2011, Math. Ann. (2013).
%H M. R. Bush and D. C. Mayer, <a href="http://arxiv.org/abs/1312.0251">3-class field towers of exact length 3</a>, Preprint: arXiv:1312.0251v1 [math.NT], 2013.
%H D. C. Mayer, <a href="https://doi.org/10.1142/S179304211250025X">The second p-class group of a number field</a>, Int. J. Number Theory 8 (2) (2012), 471-505.
%H D. C. Mayer, <a href="http://arxiv.org/abs/1403.3899">The second p-class group of a number field</a>
%H D. C. Mayer, <a href="https://doi.org/10.1007/s00605-010-0277-x">Transfers of metabelian p-groups</a>, Monatsh. Math. 166 (3-4) (2012), 467-495.
%H D. C. Mayer, <a href="http://arxiv.org/abs/1403.3896">Transfers of metabelian p-groups</a>, arXiv:1403.3896 [math.GR], 2014.
%H D. C. Mayer, <a href="https://doi.org/10.5802/jtnb.842">The distribution of second p-class groups on coclass graphs</a>, J. Théor. Nombres Bordeaux 25 (2) (2013), 401-456.
%H D. C. Mayer, <a href="http://arxiv.org/abs/1403.3833">The distribution of second p-class groups on coclass graphs</a>, arXiv:1403.3833 [math.NT], 2014.
%H D. C. Mayer, <a href="http://arxiv.org/abs/1403.3839">Principalization algorithm via class group structure</a>, Preprint: arXiv:1403.3839v1 [math.NT], 2014.
%H Daniel C. Mayer, <a href="https://arxiv.org/abs/1504.00851">Periodic sequences of p-class tower groups</a>, arXiv:1504.00851 [math.NT], 2015.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Artin_transfer_(group_theory)#Example">Artin transfer (group theory), Table 2</a>
%e 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.
%e 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.
%e a(0) and a(1) are due to D. C. Mayer (2012).
%e a(2) and a(3) are due to N. Boston, M. R. Bush and F. Hajir (2013).
%Y Cf. A242862, A242863, A242878 (supersequences), A247692, A247693, A247695, A247696, A247697 (disjoint sequences).
%K hard,more,nonn
%O 0,1
%A _Daniel Constantin Mayer_, Oct 12 2014
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