A242863 - OEIS

A242863

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

%I #26 Sep 08 2022 08:46:08

%S 3896,4027,6583,8751,9748,12067,12131,15544,16627,17131,18555,19187,

%T 19651,20276,20568,21224,21668,22395,22443,22711,23428,23683,24340,

%U 24884,24904,25447,26139,26760,27156,27355,27640

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

%C 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.

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

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

%H J. R. Brink and R. Gold, <a href="http://dx.doi.org/10.1007/BF01168670">Class field towers of imaginary quadratic fields</a>, manuscripta math. 57 (1987), 425-450.

%H M. R. Bush and D. C. Mayer, <a href="http://arxiv.org/abs/1312.0251">3-class field towers of exact length 3</a>, arXiv:1312.0251 [math.NT], 2013, J. Number Theory (2014)

%H A. Scholz and O. Taussky, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002172852&IDDOC=253437">Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper</a>, J. Reine Angew. Math. 171 (1934), 19-41.

%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>

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

%o (Magma)

%o 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;

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

%K easy,nonn

%O 1,1

%A _Daniel Constantin Mayer_, May 24 2014