A242878 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) and Hilbert 3-class field tower of exact length 3, except for the cases mentioned in the COMMENTS.

9748, 15544, 16627, 17131, 18555, 21668, 22395, 22443, 23683, 24884, 27640, 28279, 31271, 34027, 34867, 35539, 37988, 39736, 42619, 42859, 43847, 45887, 48472, 48667, 50983, 51348, 53843, 54319, 58920, 60196, 60895

Offset: 1

Author

Daniel Constantin Mayer, May 25 2014

Keywords

• hard
• nonn

Comments

CAVEAT: Up to 10^5, the length of the 3-tower is unknown for the following discriminants: 17131, 21668, 24884, 28279, 34027, 35539, 64952, 65203, 72591, 92660, 92827. The performance of the MAGMA script in section PROG would be much slower, if the class number of the first Hilbert 3-class field were computed. This would admit a criterion for the exclusion of the mentioned exceptional discriminants. Therefore, including the superfluous brushwood was the lesser of two evils.

Examples

			The case 9748 (n=1) was discussed very thoroughly by Scholz and Taussky in 1934. However, this is the famous case where they erroneously claimed that the 3-tower has exactly two stages. Brink and Gold had doubts about this claim but were unable to exclude it definitely in 1987. Bush and Mayer were the first who succeeded in disproving this claim rigorously in 2012.
		

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], J. Number Theory, accepted for publication, 2014
• A. Scholz and O. Taussky, Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper, J. Reine Angew. Math. 171 (1934), 19-41.

Crossrefs

Cf. A242862, A242863 (supersequences), and A242864, A242873 (disjoint sequences).

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 ([3, 3] eq pPrimaryInvariants(C, 3)) then E := AbelianExtension(mC); sS := Subgroups(C: Quot := [3]); sA := [AbelianExtension(Inverse(mQ)*mC) where Q, mQ := quo: x in sS]; sN := [NumberField(x): x in sA]; sF := [AbsoluteField(x): x in sN]; sM := [MaximalOrder(x): x in sF]; sM := [OptimizedRepresentation(x): x in sF]; sA := [NumberField(DefiningPolynomial(x)): x in sM]; sO := [Simplify(LLL(MaximalOrder(x))): x in sA]; delete sA, sN, sF, sM; p := 0; e := 0; for j in [1..#sO] do CO := ClassGroup(sO[j]); if (3 eq Valuation(#CO, 3)) then if ([3, 3, 3] eq pPrimaryInvariants(CO, 3)) then e := e+1; end if; else p := p+1; end if; end for; if (1 eq p) and ((0 eq e) or (1 eq e)) then d, ", "; end if; end if; end if; end for;