A247691 - OEIS

%I #18 Sep 08 2022 08:46:09

%S 3896,4027,6583,8751,12067,12131,19187,19651,20276,20568,21224,22711,

%T 23428,24340,24904,25447,26139,26760,27355,27991,28031,28759,31639,

%U 31999,32968,34088,34507,35367,36276,36807,37219,37540,39819,40299,40692,41015,41063,41583,41671,42423,43192

%N Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) whose second 3-class group is located on the sporadic part of the coclass graph G(3,2) outside of coclass trees.

%C These fields are characterized either by their 3-principalization types (transfer kernel types, TKTs) (2143), G.19, (2241), D.10, (4224), D.5, (4443), H.4, or equivalently by their transfer target types (TTTs) [(3,9)^4], [(3,3,3), (3,9)^3], [(3,3,3)^2, (3,9)^2], [(3,3,3)^3, (3,9)] (called IPADs by Boston, Bush, Hajir). The latter are used in the MAGMA PROG, which essentially constitutes the principalization algorithm via class group structure.

%H N. Boston, M. R. Bush, F. Hajir, <a href="http://arxiv.org/abs/1111.4679">Heuristics for p-class towers of imaginary quadratic fields</a>, arXiv:1111.4679 [math.NT], 2011, Math. Ann. (2013).

%H D. C. Mayer, <a href="https://arxiv.org/abs/1403.3899">The second p-class group of a number field</a>, arXiv:1403.3899 [math.NT], 2014; Int. J. Number Theory 8 (2012), no. 2, 471-505.

%H D. C. Mayer, <a href="https://arxiv.org/abs/1403.3896">Transfers of metabelian p-groups</a>, arXiv:1403.3896 [math.GR], 2014; Monatsh. Math. 166 (3-4) (2012), 467-495.

%H D. C. Mayer, <a href="https://arxiv.org/abs/1403.3833">The distribution of second p-class groups on coclass graphs</a>, arXiv:1403.3833 [math.NT], 2014; J. Théor. Nombres Bordeaux 25 (2) (2013), 401-456.

%H D. C. Mayer, <a href="http://arxiv.org/abs/1403.3839">Principalization algorithm via class group structure</a>, J. Théor. Nombres Bordeaux (2014), Preprint: arXiv:1403.3839v1 [math.NT], 2014.

%o (Magma)

%o 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; for j in [1..#sO] do CO := ClassGroup(sO[j]); if not (3 eq Valuation(#CO, 3)) then g := false; end if; end for; if (true eq g) then d, ", "; end if; end if; end if; end for;

%Y Cf. A242862, A242863 (supersequences), A242864, A242873, A247688 (subsequences), and A242878 (disjoint sequence).

%K hard,nonn

%O 1,1

%A _Daniel Constantin Mayer_, Sep 27 2014