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%I #26 Sep 08 2022 08:46:08
%S 3299,3896,4027,5703,6583,8751,9748,10015,11651,12067,12131,15544,
%T 16627,17131,17399,17723,18555,19187,19427,19651,19679,19919,20276,
%U 20568,21224,21668,22395,22443,22711,23428,23683
%N Absolute discriminants of complex quadratic fields with 3-class rank 2.
%C The length of the Hilbert 3-class field tower of a complex quadratic field is infinite for 3-class rank at least 3, and it is 1 for 3-class rank 1. In contrast, the length is at least 2 but unbounded for 3-class rank 2, whence this is the only unsolved interesting case.
%C The terms 3299, 4027 and 9748 have been discussed in detail by Scholz and Taussky. In a footnote they also mention 3896 with an erroneous claim.
%D H. Koch, B. B. Venkov, Über den p-Klassenkörperturm eines imaginär-quadratischen Zahlkörpers, Astérisque 24-25 (1975), 57-67.
%H C. McLeman, <a href="http://arxiv.org/abs/1008.3003">p-tower groups over quadratic imaginary number fields</a>, arXiv:1008.3003 [math.NT], 2010; Ann. Sci. Math. Québec 32 (2008), no. 2, 199-209.
%H A. Scholz and O. Taussky, <a href="https://eudml.org/doc/149881">Die Hauptideale der kubischen Klassenkörper imaginär-quadratischer Zahlkörper</a>, J. Reine Angew. Math. 171 (1934), 19-41. DOI:10.1515/crll.1934.171.19
%e For n=1,4, resp. n=2,3, the 3-class group is of type (3,9), resp. (3,3).
%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 (2 eq #pPrimaryInvariants(C,3)) then d,","; end if; end if; end for;
%K easy,nonn
%O 1,1
%A _Daniel Constantin Mayer_, May 24 2014
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