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
Examples
The exact length of the 3-class field tower is 2 for n=2,4,7, and 3 for n=5,8,9.
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
- 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
Crossrefs
Cf. A242862 (supersequence with arbitrary 3-class rank 2).
Programs
-
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;
Comments