A242863 Absolute discriminants of complex quadratic fields with 3-class group of elementary abelian type (3,3) of rank 2.

3896, 4027, 6583, 8751, 9748, 12067, 12131, 15544, 16627, 17131, 18555, 19187, 19651, 20276, 20568, 21224, 21668, 22395, 22443, 22711, 23428, 23683, 24340, 24884, 24904, 25447, 26139, 26760, 27156, 27355, 27640

Offset

1,1

Comments

This is the best studied subsequence of A242862. For all these discriminants, the metabelianization of the 3-tower group is known. For two extensive subsequences the 3-class tower has exact length 2, resp. 3.

References

F.-P. Heider, B. Schmithals, Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen, J. reine angew. Math. 336 (1982), 1 - 25.

B. Nebelung, Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3,3) und Anwendung auf das Kapitulationsproblem, Inauguraldissertation, Univ. zu Köln, 1989.

Links

Table of n, a(n) for n=1..31.

J. R. Brink and R. Gold, Class field towers of imaginary quadratic fields, manuscripta math. 57 (1987), 425-450.

M. R. Bush and D. C. Mayer, 3-class field towers of exact length 3, arXiv:1312.0251 [math.NT], 2013, J. Number Theory (2014)

A. Scholz and O. Taussky, Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper, J. Reine Angew. Math. 171 (1934), 19-41.

Index entries for sequences related to groups

Example

The exact length of the 3-class field tower is 2 for n=2,4,7, and 3 for n=5,8,9.

Prog

(Magma)
for d := 2 to 10^5 do a := false; if (3 eq d mod 4) and IsSquarefree(d) then a := true; end if; if (0 eq d mod 4) then r := d div 4; if IsSquarefree(r) and ((2 eq r mod 4) or (1 eq r mod 4)) then a := true; end if; end if; if (true eq a) then K := QuadraticField(-d); C := ClassGroup(K); if ([3, 3] eq pPrimaryInvariants(C, 3)) then d, ", "; end if; end if; end for;

Crossrefs

Cf. A242862 (supersequence with arbitrary 3-class rank 2).

Sequence in context: A223690 A186555 A185862 * A247691 A242873 A135202

Adjacent sequences: A242860 A242861 A242862 * A242864 A242865 A242866

Keyword

easy,nonn

Author

Daniel Constantin Mayer, May 24 2014

Status

approved