

A247694


Minimal absolute discriminants a(n) of complex quadratic fields with 3class group of type (3,3), 3principalization type H.4 (2122), 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) H.4 (2122) is not a permutation, contains a transposition, and has no fixed point.
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^{n+2},3^{n+3}),(3,3,3),(3,9)^2].
The second 3class group G is one of two vertices of depth 2 on the coclass tree with root SmallGroup(243,6) 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, A247697.
A247694 is an extremely sparse subsequence of A242878 and it is exceedingly hard to compute a(n) for n>0.
The initial term a(0)=21668 has been recognized as a realization of TKT H.4 in the Dissertation of J. R. Brink(1984). However, Brink did not know that the TKT H.4 can also occur with second 3class group G=SmallGroup(729,45) of nilpotency class cl(G)=4 and TTT [(3,3,3)^3,(3,9)]. Actually, D. C. Mayer (1991) was the first who proved that the integer 21668 is the smallest term of A247694 and does not belong to A242873.


REFERENCES

J. R. Brink, The class field tower for imaginary quadratic number fields of type (3,3), Dissertation, The Ohio State University, 1984.
D. C. Mayer, Principalization in complex S_3 fields, Congressus Numerantium 80 (1991), 7387. (Proceedings of the Twentieth Manitoba Conference on Numerical Mathematics and Computing, The University of Manitoba, Winnipeg, Manitoba, Canada, 1990.)


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
D. C. Mayer, Transfers of metabelian pgroups, Monatsh. Math. 166 (34) (2012), 467495.
D. C. Mayer, Transfers of metabelian pgroups
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
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, 2015.
Wikipedia, Artin transfer (group theory), Table 2


EXAMPLE

For a(0)=21668, we have the ground state of TKT H.4 with TTT [(9,27),(3,3,3),(3,9)^2] and cl(G)=6.
For a(1)=446788, we have the first excited state of TKT H.4 with TTT [(27,81),(3,3,3),(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, A247695, A247696, A247697 (disjoint sequences).
Sequence in context: A271021 A262818 A213870 * A235219 A232121 A232419
Adjacent sequences: A247691 A247692 A247693 * A247695 A247696 A247697


KEYWORD

hard,more,nonn


AUTHOR

Daniel Constantin Mayer, Oct 12 2014


STATUS

approved



