A242862 Absolute discriminants of complex quadratic fields with 3-class rank 2.

3299, 3896, 4027, 5703, 6583, 8751, 9748, 10015, 11651, 12067, 12131, 15544, 16627, 17131, 17399, 17723, 18555, 19187, 19427, 19651, 19679, 19919, 20276, 20568, 21224, 21668, 22395, 22443, 22711, 23428, 23683

Offset: 1

Author

Daniel Constantin Mayer, May 24 2014

Keywords

• easy
• nonn

Comments

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.

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.

Examples

			For n=1,4, resp. n=2,3, the 3-class group is of type (3,9), resp. (3,3).
		

Links

• H. Koch and B. B. Venkov, Über den p-Klassenkörperturm eines imaginär-quadratischen Zahlkörpers, Astérisque 24-25 (1975), 57-67.
• C. McLeman, p-tower groups over quadratic imaginary number fields, arXiv:1008.3003 [math.NT], 2010; Ann. Sci. Math. Québec 32 (2008), no. 2, 199-209.
• A. Scholz and O. Taussky, Die Hauptideale der kubischen Klassenkörper imaginär-quadratischer Zahlkörper, J. Reine Angew. Math. 171 (1934), 19-41. DOI:10.1515/crll.1934.171.19

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 (2 eq #pPrimaryInvariants(C,3)) then d,","; end if; end if; end for;