

A242873


Absolute discriminants of complex quadratic fields with 3class group of type (3,3), 3principalization type (4443), IPAD [(3,3,3)^3, (3,9)], and Hilbert 3class field tower of unknown length at least 3.


8



3896, 6583, 23428, 25447, 27355, 27991, 36276, 37219, 37540, 39819, 41063
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OFFSET

1,1


COMMENTS

For all these discriminants, the metabelianization of the 3tower group is the unbalanced group SmallGroup(729,45), 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.
These fields are characterized either by their 3principalization type (transfer kernel type, TKT) (4443), H.4, or equivalently by their transfer target type (TTT) [(3,3,3)^3, (3,9)] (called IPAD by Boston, Bush, Hajir). The latter is used in the MAGMA PROG. The TKT (4443) is not a permutation, contains a transposition, and has no fixed point.  Daniel Constantin Mayer, Sep 22 2014


REFERENCES

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..11.
L. Bartholdi and M. R. Bush, Maximal unramified 3extensions of imaginary quadratic fields and SL_2Z_3, J. Number Theory 124 (2007), 159166.
N. Boston, M. R. Bush, F. Hajir, Heuristics for pclass towers of imaginary quadratic fields, arXiv:1111.4679 [math.NT], 2011.
D. C. Mayer, The distribution of second pclass groups on coclass graphs, arXiv:1403.3833 [math.NT], 2014.


EXAMPLE

Already the smallest term 3896 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 (3 eq e) then d, ", "; end if; end if; end if; end for;


CROSSREFS

Cf. A242862, A242863 (supersequences), and A242864, A242878 (disjoint sequences).
Sequence in context: A185862 A242863 A247691 * A135202 A204147 A252138
Adjacent sequences: A242870 A242871 A242872 * A242874 A242875 A242876


KEYWORD

hard,nonn


AUTHOR

Daniel Constantin Mayer, May 24 2014


EXTENSIONS

Definition completed by Daniel Constantin Mayer, Sep 22 2014


STATUS

approved



