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A250240
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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).
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7
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2177, 2677, 4841, 6289, 6940, 6997, 8789, 9869, 11324, 17448, 17581, 23192, 23417, 24433, 25741, 26933, 30273, 33765, 34253, 34412, 34968, 35537, 36376, 38037, 38057, 40773, 41224, 42152, 42649, 43176, 43349, 44617, 45529, 47528
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OFFSET
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1,1
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COMMENTS
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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.
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.
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.
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
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
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
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REFERENCES
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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.
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
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LINKS
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PROG
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(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;
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CROSSREFS
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KEYWORD
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hard,more,nonn
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AUTHOR
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STATUS
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approved
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