OFFSET
1,1
COMMENTS
For the discriminants d in A250239, the 3-class field tower of K=Q(sqrt(-3),sqrt(d)) has exactly two stages and the second 3-class group G of K is given by the metabelian 3-group G=SmallGroup(729,95) with transfer kernel type a.1, (0,0,0,0), transfer target type [(9,27),(3,3)^3] and coclass 1. This is the first excited state on the coclass-1 graph.
The reason the 3-class field tower of K must stop at the second Hilbert 3-class field is Blackburn's Theorem on two-generated 3-groups G whose commutator subgroup G' also has two generators. In fact, the group G=SmallGroup(729,95) has commutator subgroup G'=(9,9), two-generated.
REFERENCES
H. U. Besche, B. Eick, and E. A. O'Brien, The SmallGroups Library - a Library of Groups of Small Order, 2005, an accepted and refereed GAP 4 package, available also in MAGMA.
LINKS
N. Blackburn, On prime-power groups in which the derived group has two generators, Proc. Camb. Phil. Soc. 53 (1957), 19-27.
D. C. Mayer, The second p-class group of a number field, Int. J. Number Theory 8 (2) (2012), 471-505.
D. C. Mayer, The second p-class group of a number field. Preprint: arXiv:1403.3899v1 [math.NT], 2014.
D. C. Mayer, Principalization algorithm via class group structure, Preprint: arXiv:1403.3839v1 [math.NT], 2014. J. Théor. Nombres Bordeaux 26 (2014), no. 2, 415-464.
PROG
(Magma) SetClassGroupBounds("GRH"); for n := 7453 to 20000 do cnd := false; if (1 eq n mod 4) and IsSquarefree(n) then cnd := true; end if; if (0 eq n mod 4) then r := n div 4; if IsSquarefree(r) and ((2 eq r mod 4) or (3 eq r mod 4)) then cnd := true; end if; end if; if (true eq cnd) then R := QuadraticField(n); E := QuadraticField(-3); K := Compositum(R, E); C, mC := ClassGroup(K); if ([3, 3] eq pPrimaryInvariants(C, 3)) then s := Subgroups(C: Quot := [3]); a := [AbelianExtension(Inverse(mq)*mC) where _, mq := quo<C|x`subgroup> : x in s]; b := [NumberField(x) : x in a]; d := [MaximalOrder(x) : x in a]; b := [AbsoluteField(x) : x in b]; c := [MaximalOrder(x) : x in b]; c := [OptimizedRepresentation(x) : x in b]; b := [NumberField(DefiningPolynomial(x)) : x in c]; a := [Simplify(LLL(MaximalOrder(x))) : x in b]; if IsNormal(b[2]) then H := Compositum(NumberField(a[1]), NumberField(a[2])); else H := Compositum(NumberField(a[1]), NumberField(a[3])); end if; O := MaximalOrder(H); CH := ClassGroup(LLL(O)); if ([9, 9] eq pPrimaryInvariants(CH, 3)) then n, ", "; end if; end if; end if; end for;
CROSSREFS
KEYWORD
hard,more,nonn
AUTHOR
Daniel Constantin Mayer, Nov 16 2014
STATUS
approved