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

1,1

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.

References

H. Koch, B. B. Venkov, Über den p-Klassenkörperturm eines imaginär-quadratischen Zahlkörpers, Astérisque 24-25 (1975), 57-67.

Links

Table of n, a(n) for n=1..31.

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

Example

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

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

Crossrefs

Sequence in context: A215565 A068755 A330253 * A078951 A236660 A106721

Adjacent sequences: A242859 A242860 A242861 * A242863 A242864 A242865

Keyword

easy,nonn

Author

Daniel Constantin Mayer, May 24 2014

Status

approved