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.

17131, 819743, 2244399, 30224744

Offset

0,1

Comments

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.

Links

Table of n, a(n) for n=0..3.

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

Example

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).

Crossrefs

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

Sequence in context: A233610 A361349 A121669 * A218247 A235891 A251207

Adjacent sequences: A247694 A247695 A247696 * A247698 A247699 A247700

Keyword

hard,more,nonn

Author

Daniel Constantin Mayer, Oct 12 2014

Status

approved