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A247692 Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type E.6 (1122), and second 3-class group G of odd nilpotency class cl(G)=2(n+2)+1. 6
15544, 268040, 1062708, 27629107 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The 3-principalization type (transfer kernel type, TKT) E.6 (1122) is not a permutation and has a single fixed point.

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 a vertex 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.

A247692 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.

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

Wikipedia, Artin transfer (group theory), Table 2

EXAMPLE

For a(0)=15544, we have the ground state of TKT E.6 with TTT [(9,27),(3,3,3),(3,9)^2] and cl(G)=5.

For a(1)=268040, we have the first excited state of TKT E.6 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), A247693, A247694, A247695, A247696, A247697 (disjoint sequences).

Sequence in context: A201344 A235278 A100973 * A262793 A206104 A205654

Adjacent sequences:  A247689 A247690 A247691 * A247693 A247694 A247695

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.