A247688 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3), 3-principalization type (2143), IPAD [(3,9)^4], and Hilbert 3-class field tower of unknown length at least 3.

12067, 49924, 54195, 60099, 83395, 86551, 91643, 93067, 96551

Offset: 1

Author

Daniel Constantin Mayer, Sep 22 2014

Keywords

• hard
• more
• nonn

Comments

These fields are characterized either by their 3-principalization type (transfer kernel type, TKT) (2143), G.19, or equivalently by their transfer target type (TTT) [(3,9)^4] (called IPAD by Boston, Bush, Hajir). The latter is used in the MAGMA PROG. The TKT (2143) is a permutation composed of two disjoint transpositions without fixed point.

For all these discriminants, the metabelianization of the 3-tower group is the unbalanced group SmallGroup(729, 57), whence it is completely open whether the tower must terminate at a finite stage or not. Consequently, these discriminants are among the foremost challenges of future research.

12067 has been discovered by Heider and Schmithals.

Examples

			Already the smallest term 12067 resists all attempts to determine the length of its Hilbert 3-class field tower.
		

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, arXiv:1111.4679 [math.NT], 2011, Math. Ann. (2013).
• D. C. Mayer, The distribution of second p-class groups on coclass graphs, arXiv:1403.3833 [math.NT], 2014.

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, 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 (0 eq e) then d, ", "; end if; end if; end if; end for;