A247697 - OEIS

%I #23 Aug 06 2020 04:24:07

%S 17131,819743,2244399,30224744

%N 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.

%C The 3-principalization type (transfer kernel type, TKT) G.16 (2134) is a permutation, contains a transposition, and has two fixed points.

%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 [(3,9),(3^{n+2},3^{n+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(243,8) 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, A247694.

%C A247697 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 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>, arXiv:1403.3899 [math.NT], 2014.

%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)=17131, we have the ground state of TKT G.16 with TTT [(3,9),(9,27),(3,9)^2] and cl(G)=6.

%e 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.

%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, A247694, A247695, A247696 (disjoint sequences).

%K hard,more,nonn

%O 0,1

%A _Daniel Constantin Mayer_, Oct 12 2014