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%I #29 Sep 08 2022 08:46:10
%S 2177,2677,4841,6289,6940,6997,8789,9869,11324,17448,17581,23192,
%T 23417,24433,25741,26933,30273,33765,34253,34412,34968,35537,36376,
%U 38037,38057,40773,41224,42152,42649,43176,43349,44617,45529,47528
%N Fundamental discriminants d uniquely characterizing all complex biquadratic fields Q(sqrt(-3),sqrt(d)) which have 3-class group of type (3,3) and second 3-class group isomorphic to SmallGroup(729,37).
%C For the discriminants d in A250240, the 3-class field tower of K=Q(sqrt(-3),sqrt(d)) has at least three stages and the second 3-class group G of K is given by G=SmallGroup(729,37), which is called the non-CF group A by Ascione, Havas and Leedham-Green. It has many properties (transfer kernel type b.10, (0,0,4,3), and transfer target type [(3,9)^2,(3,3,3)^2]) coinciding with those of SmallGroup(729,34), called the non-CF group H. Both are immediate descendants of SmallGroup(243,3) and can only be distinguished by their commutator subgroup G', which is of type (3,3,9) for A, and (3,3,3,3) for H.
%C Since the verification of the structure of G' requires computation of the 3-class group of the Hilbert 3-class field of K, which is of absolute degree 36 over Q, the construction of A250240 is extremely tough.
%C Whereas the metabelian 3-group A is rather well behaved, possessing six terminal immediate descendants only, the notorious group H is famous for giving rise to three infinite coclass trees with non-metabelian mainlines and horrible complexity.
%C In 66.2 hours of CPU time, Magma computed all 34 discriminants d up to the bound 50000. Starting with d=38057, Magma begins to struggle considerably, since an increasing amount of time (NOT included above) is used for swapping to the hard disk. - _Daniel Constantin Mayer_, Dec 02 2014
%C The given Magma PROG works correctly up to 10000. However, for ranges beyond 10000, a complication arises, since the non-CF group B = SmallGroup(729,40) also has a commutator subgroup of type (3,3,9) and must be sifted with the aid of its different transfer target type [(9,9),(3,9),(3,3,3)^2]. Up to 50000, this occurs three times for d in {17609,30941,31516}. - _Daniel Constantin Mayer_, Dec 05 2014
%C The group G=SmallGroup(729,37) has p-multiplicator rank m(G)=5. By Theorem 6 of I. R. Shafarevich (with misprint corrected) the relation rank of the 3-class tower group H is bounded by r(H) <= d(H) + r + 1 = 2 + 1 + 1 = 4, where d(H) denotes the generator rank of H and r is the torsionfree unit rank of K. Thus, G with r(G) >= m(G) = 5 cannot be the 3-class tower group of K and the tower must have at least three stages. - _Daniel Constantin Mayer_, Sep 24 2015
%D 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.
%D I. R. Shafarevich, Extensions with prescribed ramification points, Publ. Math., Inst. Hautes Études Sci. 18 (1964), 71-95 (Russian). English transl. by J. W. S. Cassels: Am. Math. Soc. Transl., II. Ser., 59 (1966), 128-149. - _Daniel Constantin Mayer_, Sep 24 2015
%H J. A. Ascione, G. Havas, and C. R. Leedham-Green, <a href="http://dx.doi.org/10.1017/S0004972700010467">A computer aided classification of certain groups of prime power order</a>, Bull. Austral. Math. Soc. 17 (1977), 257-274.
%H D. C. Mayer, <a href="http://www.worldscientific.com/doi/abs/10.1142/S179304211250025X">The second p-class group of a number field</a>, Int. J. Number Theory 8 (2) (2012), 471-505.
%H D. C. Mayer, <a href="http://arxiv.org/abs/1403.3899">The second p-class group of a number field</a>. Preprint: arXiv:1403.3899v1 [math.NT], 2014.
%H D. C. Mayer, <a href="http://arxiv.org/abs/1403.3839">Principalization algorithm via class group structure</a>, Preprint: arXiv:1403.3839v1 [math.NT], 2014. J. Théor. Nombres Bordeaux 26 (2014), no. 2, 415-464.
%o (Magma)SetClassGroupBounds("GRH"); for n := 2177 to 10000 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 ([3,3,9] eq pPrimaryInvariants(CH, 3)) then n, ", "; end if; end if; end if; end for;
%Y A006832, A250235, A250236 are supersequences.
%Y A250237, A250238, A250239,A250241, A250242 are disjoint sequences.
%K hard,more,nonn
%O 1,1
%A _Daniel Constantin Mayer_, Nov 15 2014
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