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A250241
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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(729,34).
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7
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2589, 4853, 7881, 8057, 8769, 9905, 11697, 20693, 21281, 21337, 24917, 25185, 27548, 28061, 28137, 28936, 28940, 29485, 33864, 35224, 37916, 39633, 41628, 49461, 49541
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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 A250241, 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(729,34), which is called the non-CF group H by Ascione, Havas and Leedham-Green. It has properties very similar to those of SmallGroup(729,37), called the non-CF group A. Both are immediate descendants of SmallGroup(243,3) and can only be distinguished by their commutator subgroup G', which is of type (3,3,3,3) for H, and (3,3,9) for A.
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 A250241 is extremely tough.
In 40.5 hours of CPU time, Magma computed all 25 discriminants d up to the bound 50000. Starting with d=37916, Magma begins to struggle considerably, since an increasing amount of time (NOT included above) is used for swapping to the hard disk. A very powerful machine would be required for continuing beyond 50000. - Daniel Constantin Mayer, Dec 02 2014
The group G=SmallGroup(729,34) 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 := 2589 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, 3, 3] 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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