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A242863 Absolute discriminants of complex quadratic fields with 3-class group of elementary abelian type (3,3) of rank 2. 16
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 (list; graph; refs; listen; history; text; internal format)
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
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.
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
KEYWORD
easy,nonn
AUTHOR
STATUS
approved

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Last modified July 23 17:49 EDT 2024. Contains 374553 sequences. (Running on oeis4.)