A247690 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) and 3-principalization type (4224).

12131, 19187, 20276, 20568, 24340, 26760, 31639, 31999, 32968, 34507, 35367, 41583, 41671, 43307, 57079, 64196, 73731, 85796, 87720, 93823, 95691

Offset: 1

Author

Daniel Constantin Mayer, Sep 23 2014

Keywords

• hard
• more
• nonn

Comments

These fields are characterized either by their 3-principalization type (transfer kernel type, TKT) (4224), D.5, or equivalently by their transfer target type (TTT) [(3,3,3)^2, (3,9)^2] (called IPAD by Boston, Bush, Hajir). The latter is used in the MAGMA PROG, which essentially constitutes the principalization algorithm via class group structure. The TKT (4224) has two fixed points and is not a permutation.

For all these discriminants, the 3-tower group is the metabelian Schur sigma-group SmallGroup(243, 7) and the Hilbert 3-class field tower terminates at the second stage.

12131 has been discovered by Heider and Schmithals.

References

• F.-P. Heider, B. Schmithals, Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen, J. reine angew. Math. 336 (1982), 1 - 25.
• D. C. Mayer, "The distribution of second p-class groups on coclass graphs", J. Théor. Nombres Bordeaux 25 (2) (2013), 401-456.

Links

• 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, Principalization algorithm via class group structure, J. Théor. Nombres Bordeaux (2014), Preprint: arXiv:1403.3839v1 [math.NT], 2014.

Crossrefs

Cf. A242862, A242863, A242864 (supersequences), and A247689, 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, 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 (2 eq e) then d, ", "; end if; end if; end if; end for;