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A247693 Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type E.14 (3122), and second 3-class group G of odd nilpotency class cl(G)=2(n+2)+1. 6
16627, 262744, 4776071, 40059363 (list; graph; refs; listen; history; text; internal format)
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.

A247693 is an extremely sparse subsequence of A242878 and it is exceedingly hard to compute a(n) for n>0.

REFERENCES

D. C. Mayer, The second p-class group of a number field, Int. J. Number Theory 8 (2) (2012), 471-505.

D. C. Mayer, Transfers of metabelian p-groups, Monatsh. Math. 166 (3-4) (2012), 467-495.

D. C. Mayer, "The distribution of second p-class groups on coclass graphs", J. Théor. Nombres Bordeaux 25 (2) (2013), 401-456.

LINKS

Table of n, a(n) for n=0..3.

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, 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

Cf. A242862, A242863, A242878 (supersequences), A247692, A247694, A247695, A247696, A247697 (disjoint sequences).

Sequence in context: A250930 A210272 A183657 * A170788 A057329 A232450

Adjacent sequences:  A247690 A247691 A247692 * A247694 A247695 A247696

KEYWORD

hard,more,nonn

AUTHOR

Daniel Constantin Mayer, Sep 28 2014

STATUS

approved

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Last modified January 17 07:18 EST 2018. Contains 297787 sequences.