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

1,1

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.

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

Table of n, a(n) for n=1..9.

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.

Example

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

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, 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<C|x`subgroup>: 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;

Crossrefs

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

Sequence in context: A292815 A252304 A251680 * A236726 A236701 A259523

Adjacent sequences: A247685 A247686 A247687 * A247689 A247690 A247691

Keyword

hard,more,nonn

Author

Daniel Constantin Mayer, Sep 22 2014

Status

approved