

A247696


Minimal absolute discriminants a(n) of complex quadratic fields with 3class group of type (3,3), 3principalization type E.9 (2334), 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.9 (2334) is not a permutation and has two fixed points.
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 [(3,9),(3^{n+2},3^{n+3}),(3,9)^2].
The second 3class 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).
All these fields possess a Hilbert 3class field tower of exact length 3.
A247696 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..4.
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)=9748, we have the ground state of TKT E.9 with TTT [(3,9),(9,27),(3,9)^2] and cl(G)=5.
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.
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.
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.
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.
a(0) and a(1) are due to D. C. Mayer (2012).
a(2), a(3) and a(4) are due to N. Boston, M. R. Bush and F. Hajir (2013).


CROSSREFS

Cf. A242862, A242863, A242878 (supersequences), A247692, A247693, A247694, A247695, A247697 (disjoint sequences).
Sequence in context: A023687 A204286 A242878 * A010092 A023339 A145209
Adjacent sequences: A247693 A247694 A247695 * A247697 A247698 A247699


KEYWORD

hard,more,nonn


AUTHOR

Daniel Constantin Mayer, Sep 28 2014


STATUS

approved



