The OEIS is supported by the many generous donors to the OEIS Foundation.


(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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)
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.
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.
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).
Cf. A242862, A242863, A242878 (supersequences), A247693, A247694, A247695, A247696, A247697 (disjoint sequences).
Sequence in context: A235278 A329769 A100973 * A262793 A206104 A205654

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 30 05:35 EDT 2023. Contains 365781 sequences. (Running on oeis4.)