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A250242
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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 either SmallGroup(2187,247)-#1;5 or SmallGroup(2187,247)-#1;9.
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7
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11608, 14056, 20521, 21109, 25949, 27245, 27329, 31065, 32421, 32765, 38085, 38285, 39853, 40156, 43257, 45541, 46489, 48481
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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 A250242, 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 of coclass 2, given by either SmallGroup(2187,247)-#1;5 or SmallGroup(2187,247)-#1;9. (Note that these groups G are of order 6561 and lie outside of the SmallGroups Library, whence we must use the terminology for descendants defined in the ANUPQ package of Magma and GAP.) Both have transfer kernel type b.10, (0,0,4,3), and full transfer target type [(3,3);(9,27),(3,9),(3,3,3)^2;(3,9,27)]). Both are immediate descendants of the mainline group SmallGroup(2187,247) on the coclass tree with root SmallGroup(729,40) and cannot be distinguished by any known arithmetical criteria. Their commutator subgroup G' is of type (3,9,27).
Since the verification of the structure of G' (used by the given Magma PROG) 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 A250242 is rather expensive.
Both groups G=SmallGroup(2187,247)-#1;5 and G=SmallGroup(2187,247)-#1;9 have 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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EXAMPLE
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Up to 50000, the discriminants 20521 and 40156 are the only two terms which show a twisted bipolarization. All the other discriminants, starting with 11608, 14056, 21109, etc., reveal the (usual) parallel bipolarization among the four unramified cyclic cubic extensions. In the twisted case, the Hilbert 3-class field of the complex quadratic subfield Q(sqrt(-3d)) gives rise to the distinguished extension of type (9,27) (contained in the transfer target type), whereas in the parallel case the Hilbert 3-class field of the real quadratic subfield Q(sqrt(d)) is responsible for (9,27).
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PROG
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(Magma) SetClassGroupBounds("GRH"); for n := 11608 to 50000 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, 9, 27] 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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nonn,hard,more
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AUTHOR
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STATUS
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approved
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