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A247696 Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type E.9 (2334), and second 3-class group G of odd nilpotency class cl(G)=2(n+2)+1. 6
9748, 297079, 1088808, 11091140, 94880548 (list; graph; refs; listen; history; text; internal format)



The 3-principalization 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 3-class 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 3-class 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 3-class 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.


Table of n, a(n) for n=0..4.

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.

Daniel C. Mayer, Periodic sequences of p-class tower groups, arXiv:1504.00851, 2015.

Wikipedia, Artin transfer (group theory), Table 2


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).


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




Daniel Constantin Mayer, Sep 28 2014



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