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%I #15 Dec 08 2018 03:02:26
%S 9748,297079,1088808,11091140,94880548
%N Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type E.9 (2334), and second 3-class group G of odd nilpotency class cl(G)=2(n+2)+1.
%C The 3-principalization type (transfer kernel type, TKT) E.9 (2334) is not a permutation and has two fixed points.
%C 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].
%C The second 3-class group G is one of two vertices of depth 1 on the coclass tree with root SmallGroup(243,8) contained in the coclass graph G(3,2).
%C All these fields possess a Hilbert 3-class field tower of exact length 3.
%C A247696 is an extremely sparse subsequence of A242878 and it is exceedingly hard to compute a(n) for n>0.
%H N. Boston, M. R. Bush, F. Hajir, <a href="http://arxiv.org/abs/1111.4679">Heuristics for p-class towers of imaginary quadratic fields</a>, Math. Ann. (2013), Preprint: arXiv:1111.4679v1 [math.NT], 2011.
%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>, J. Number Theory (2014), Preprint: arXiv:1312.0251v1 [math.NT], 2013.
%H D. C. Mayer, <a href="https://arxiv.org/abs/1403.3899">The second p-class group of a number field</a>, arXiv:1403.3899 [math.NT], 2014; Int. J. Number Theory 8 (2012), no. 2, 471-505.
%H D. C. Mayer, <a href="https://arxiv.org/abs/1403.3896">Transfers of metabelian p-groups</a>, arXiv:1403.3896 [math.GR], 2014; Monatsh. Math. 166 (3-4) (2012), 467-495.
%H D. C. Mayer, <a href="https://arxiv.org/abs/1403.3833">The distribution of second p-class groups on coclass graphs</a>, arXiv:1403.3833 [math.NT], 2014; J. Théor. Nombres Bordeaux 25 (2) (2013), 401-456.
%H D. C. Mayer, <a href="http://arxiv.org/abs/1403.3839">Principalization algorithm via class group structure</a>, J. Théor. Nombres Bordeaux (2014), Preprint: arXiv:1403.3839v1 [math.NT], 2014.
%H Daniel C. Mayer, <a href="http://arxiv.org/pdf/1504.00851.pdf">Periodic sequences of p-class tower groups</a>, arXiv:1504.00851, 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)=9748, we have the ground state of TKT E.9 with TTT [(3,9),(9,27),(3,9)^2] and cl(G)=5.
%e For a(1)=297079, we have the first excited state of TKT E.9 with TTT [(3,9),(27,81),(3,9)^2] and cl(G)=7.
%e For a(2)=1088808, we have the second excited state of TKT E.9 with TTT [(3,9),(81,243),(3,9)^2] and cl(G)=9.
%e For a(3)=11091140, we have the third excited state of TKT E.9 with TTT [(3,9),(243,729),(3,9)^2] and cl(G)=11.
%e For a(4)=94880548, we have the fourth excited state of TKT E.9 with TTT [(3,9),(729,2187),(3,9)^2] and cl(G)=13.
%e a(0) and a(1) are due to D. C. Mayer (2012).
%e a(2), a(3) and a(4) are due to N. Boston, M. R. Bush and F. Hajir (2013).
%Y Cf. A242862, A242863, A242878 (supersequences), A247692, A247693, A247694, A247695, A247697 (disjoint sequences).
%K hard,more,nonn
%O 0,1
%A _Daniel Constantin Mayer_, Sep 28 2014
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