A242873 Absolute discriminants of complex quadratic fields with 3-class group of type (3,3), 3-principalization type (4443), IPAD [(3,3,3)^3, (3,9)], and Hilbert 3-class field tower of unknown length at least 3.

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

Offset: 1

Author

Daniel Constantin Mayer, May 24 2014

Keywords

• hard
• nonn

Comments

For all these discriminants, the metabelianization of the 3-tower 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 3-principalization 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

Examples

			Already the smallest term 3896 resists all attempts to determine the length of its Hilbert 3-class field tower.
		

References

• D. C. Mayer, The distribution of second p-class groups on coclass graphs, J. Théor. Nombres Bordeaux 25 (2) (2013), 401-456.

Links

• L. Bartholdi and M. R. Bush, Maximal unramified 3-extensions of imaginary quadratic fields and SL_2Z_3, J. Number Theory 124 (2007), 159-166.
• N. Boston, M. R. Bush, F. Hajir, Heuristics for p-class towers of imaginary quadratic fields, arXiv:1111.4679 [math.NT], 2011.
• 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, A242878 (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 (3 eq e) then d, ", "; end if; end if; end if; end for;

Extensions

Definition completed by Daniel Constantin Mayer, Sep 22 2014