

A247692


Minimal absolute discriminants a(n) of complex quadratic fields with 3class group of type (3,3), 3principalization type E.6 (1122), and second 3class group G of odd nilpotency class cl(G)=2(n+2)+1.


6




OFFSET

0,1


COMMENTS

The 3principalization type (transfer kernel type, TKT) E.6 (1122) is not a permutation and has a single fixed point.
The nilpotency condition cl(G)=2n+5 for the second 3class 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].
The second 3class group G is a vertex of depth 1 on the coclass tree with root SmallGroup(243,6) contained in the coclass graph G(3,2).
All these fields possess a Hilbert 3class field tower of exact length 3.
A247692 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, F. Hajir, Heuristics for pclass towers of imaginary quadratic fields, Math. Ann. (2013), Preprint: arXiv:1111.4679v1 [math.NT], 2011.
M. R. Bush and D. C. Mayer, 3class field towers of exact length 3, J. Number Theory (2014), Preprint: arXiv:1312.0251v1 [math.NT], 2013.
D. C. Mayer, The second pclass group of a number field, arXiv:1403.3899 [math.NT], 2014; Int. J. Number Theory 8 (2012), no. 2, 471505.
D. C. Mayer, Transfers of metabelian pgroups, arXiv:1403.3896 [math.GR], 2014; Monatsh. Math. 166 (34) (2012), 467495.
D. C. Mayer, The distribution of second pclass groups on coclass graphs, arXiv:1403.3833 [math.NT], 2014; J. Théor. Nombres Bordeaux 25 (2) (2013), 401456.
D. C. Mayer, Principalization algorithm via class group structure, J. Théor. Nombres Bordeaux (2014), Preprint: arXiv:1403.3839v1 [math.NT], 2014.
Daniel C. Mayer, Periodic sequences of pclass tower groups, arXiv:1504.00851, 2015.
Wikipedia, Artin transfer (group theory), Table 2


EXAMPLE

For a(0)=15544, we have the ground state of TKT E.6 with TTT [(9,27),(3,3,3),(3,9)^2] and cl(G)=5.
For a(1)=268040, we have the first excited state of TKT E.6 with TTT [(27,81),(3,3,3),(3,9)^2] and cl(G)=7.
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), A247693, A247694, A247695, A247696, A247697 (disjoint sequences).
Sequence in context: A201344 A235278 A100973 * A262793 A206104 A205654
Adjacent sequences: A247689 A247690 A247691 * A247693 A247694 A247695


KEYWORD

hard,more,nonn


AUTHOR

Daniel Constantin Mayer, Sep 28 2014


STATUS

approved



