A242864 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) and Hilbert 3-class field tower of exact length 2.

4027, 8751, 12131, 19187, 19651, 20276, 20568, 21224, 22711, 24340, 24904, 26139, 26760, 28031, 28759, 31639, 31999, 32968, 34088, 34507, 35367, 36807, 40299, 40692, 41015, 41583, 41671, 42423, 43192, 43307, 44004

Offset: 1

Author

Daniel Constantin Mayer, May 24 2014

Keywords

• hard
• nonn

Comments

For all these discriminants, the metabelianization of the 3-tower group is one of the two Schur sigma-groups SmallGroup(243, 5) or SmallGroup(243, 7), whence it is clear that the tower must terminate at the second stage.

n = 1 is discussed very thoroughly by Scholz and Taussky.

These fields are characterized either by their 3-principalization types (transfer kernel types, TKTs) (2241), D.10, resp. (4224), D.5, or equivalently by their transfer target types (TTTs) [(3,3,3), (3,9)^3], resp. [(3,3,3)^2, (3,9)^2] (called IPADs by Boston, Bush, Hajir). The latter are used in the MAGMA PROG, which essentially constitutes the principalization algorithm via class group structure. - Daniel Constantin Mayer, Sep 23 2014

Links

• Laurent Bartholdi and Michael R. Bush, Maximal unramified 3-extensions of imaginary quadratic fields and SL_2Z_3, J. Number Theory, 124 (2007), 159-166.
• N. Boston, M. R. Bush, F. Hajir, Heuristics for p-class towers of imaginary quadratic fields, Math. Ann. (2013), Preprint: arXiv:1111.4679v1 [math.NT], 2011.
• D. C. Mayer, The distribution of second p-class groups on coclass graphs, J. Théor. Nombres Bordeaux 25 (2) (2013), 401-456.
• D. C. Mayer, Principalization algorithm via class group structure, J. Théor. Nombres Bordeaux (2014), Preprint: arXiv:1403.3839v1 [math.NT], 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), A247689, A247690 (subsequences), and A242873, A242878 (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,mC := 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; g := true; 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 g := false; end if; end for; if (true eq g) and ((1 eq e) or (2 eq e)) then d,","; end if; end if; end if; end for;