OFFSET
0,1
COMMENTS
The 3-principalization type (transfer kernel type, TKT) E.14 (3122) is not a permutation, contains a 3-cycle, and has no fixed points.
The nilpotency condition cl(G)=2n+5 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^{n+2},3^{n+3}),(3,3,3),(3,9)^2].
The second 3-class group G is one of two vertices 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 3-class field tower of exact length 3.
LINKS
N. Boston, M. R. Bush, F. Hajir, Heuristics for p-class towers of imaginary quadratic fields, Math. Ann. (2013), Preprint: arXiv:1111.4679v1 [math.NT], 2011.
M. R. Bush and D. C. Mayer, 3-class field towers of exact length 3, J. Number Theory (2014), Preprint: arXiv:1312.0251v1 [math.NT], 2013.
D. C. Mayer, The second p-class group of a number field, arXiv:1403.3899 [math.NT], 2014; Int. J. Number Theory 8 (2012), no. 2, 471-505.
D. C. Mayer, Transfers of metabelian p-groups, arXiv:1403.3896 [math.GR], 2014; Monatsh. Math. 166 (3-4) (2012), 467-495.
D. C. Mayer, The distribution of second p-class groups on coclass graphs, arXiv:1403.3833 [math.NT], 2014; J. Théor. Nombres Bordeaux 25 (2) (2013), 401-456.
D. C. Mayer, Principalization algorithm via class group structure, J. Théor. Nombres Bordeaux (2014), Preprint: arXiv:1403.3839v1 [math.NT], 2014
Wikipedia, Artin transfer (group theory), Table 2
Daniel C. Mayer, Periodic sequences of p-class tower groups, arXiv:1504.00851, 2015.
EXAMPLE
For a(0)=16627, we have the ground state of TKT E.14 with TTT [(9,27),(3,3,3),(3,9)^2] and cl(G)=5.
For a(1)=262744, we have the first excited state of TKT E.14 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
KEYWORD
hard,more,nonn
AUTHOR
Daniel Constantin Mayer, Sep 28 2014
STATUS
approved