This site is supported by donations to The OEIS Foundation.



Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 8
3896, 6583, 23428, 25447, 27355, 27991, 36276, 37219, 37540, 39819, 41063 (list; graph; refs; listen; history; text; internal format)



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


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


Table of n, a(n) for n=1..11.

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.


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



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; 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;


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




Daniel Constantin Mayer, May 24 2014


Definition completed by Daniel Constantin Mayer, Sep 22 2014



Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified November 20 13:59 EST 2017. Contains 294972 sequences.