

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. Théor. Nombres Bordeaux 25 (2) (2013), 401456.


LINKS



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



KEYWORD

hard,nonn


AUTHOR



EXTENSIONS



STATUS

approved



