A247695 Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type E.8 (2234), and second 3-class group G of odd nilpotency class cl(G)=2(n+2)+1.

34867, 370740, 4087295, 19027947

Offset

0,1

Comments

The 3-principalization type (transfer kernel type, TKT) E.8 (2234) is not a permutation and has three 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 a vertex 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.

A247695 is an extremely sparse subsequence of A242878 and it is exceedingly hard to compute a(n) for n>0.

Links

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

N. Boston, M. R. Bush and F. Hajir, Heuristics for p-class towers of imaginary quadratic fields, Preprint: arXiv:1111.4679v1 [math.NT], 2011, Math. Ann. (2013).

M. R. Bush and D. C. Mayer, 3-class field towers of exact length 3, Preprint: arXiv:1312.0251v1 [math.NT], 2013.

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

D. C. Mayer, The second p-class group of a number field, arXiv:1403.3899 [math.NT], 2014.

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

D. C. Mayer, Transfers of metabelian p-groups, arXiv:1403.3896 [math.GR], 2014.

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

D. C. Mayer, The distribution of second p-class groups on coclass graphs, arXiv:1403.3833 [math.NT], 2014.

D. C. Mayer, Principalization algorithm via class group structure, Preprint: arXiv:1403.3839v1 [math.NT], 2014.

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

Wikipedia, Artin transfer (group theory), Table 2

Example

For a(0)=34867, we have the ground state of TKT E.8 with TTT [(3,9),(9,27),(3,9)^2] and cl(G)=5.
For a(1)=370740, we have the first excited state of TKT E.8 with TTT [(3,9),(27,81),(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, A247693, A247694, A247696, A247697 (disjoint sequences).

Sequence in context: A236591 A236727 A233815 * A204886 A257755 A230976

Adjacent sequences: A247692 A247693 A247694 * A247696 A247697 A247698

Keyword

hard,more,nonn

Author

Daniel Constantin Mayer, Sep 28 2014

Status

approved