

A247688


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


3




OFFSET

1,1


COMMENTS

These fields are characterized either by their 3principalization 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 3tower 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 pclass groups on coclass graphs, J. ThÃ©or. Nombres Bordeaux 25 (2) (2013), 401456.


LINKS

Table of n, a(n) for n=1..9.
N. Boston, M. R. Bush, F. Hajir, Heuristics for pclass towers of imaginary quadratic fields, arXiv:1111.4679 [math.NT], 2011, Math. Ann. (2013).
D. C. Mayer, The distribution of second pclass 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 3class 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<Cx`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: A064966 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



