

A247697


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


6




OFFSET

0,1


COMMENTS

The 3principalization 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 3class 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 3class 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 3class 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 pclass towers of imaginary quadratic fields, Preprint: arXiv:1111.4679v1 [math.NT], 2011; Math. Ann. (2013).
M. R. Bush and D. C. Mayer, 3class field towers of exact length 3, Preprint: arXiv:1312.0251v1 [math.NT], 2013.
D. C. Mayer, The second pclass group of a number field, Int. J. Number Theory 8 (2) (2012), 471505.
D. C. Mayer, The second pclass group of a number field, arXiv:1403.3899 [math.NT], 2014.
D. C. Mayer, Transfers of metabelian pgroups, Monatsh. Math. 166 (34) (2012), 467495.
D. C. Mayer, Transfers of metabelian pgroups, arXiv:1403.3896 [math.GR], 2014.
D. C. Mayer, The distribution of second pclass groups on coclass graphs, J. ThÃ©or. Nombres Bordeaux 25 (2) (2013), 401456.
D. C. Mayer, The distribution of second pclass 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 pclass 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: A233611 A233610 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



