

A250240


Fundamental discriminants d uniquely characterizing all complex biquadratic fields Q(sqrt(3),sqrt(d)) which have 3class group of type (3,3) and second 3class group isomorphic to SmallGroup(729,37).


7



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

1,1


COMMENTS

For the discriminants d in A250240, the 3class field tower of K=Q(sqrt(3),sqrt(d)) has at least three stages and the second 3class group G of K is given by G=SmallGroup(729,37), which is called the nonCF group A by Ascione, Havas and LeedhamGreen. 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 nonCF 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 3class group of the Hilbert 3class field of K, which is of absolute degree 36 over Q, the construction of A250240 is extremely tough.
Whereas the metabelian 3group 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 nonmetabelian 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 nonCF 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 pmultiplicator rank m(G)=5. By Theorem 6 of I. R. Shafarevich (with misprint corrected) the relation rank of the 3class 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 3class tower group of K and the tower must have at least three stages.  Daniel Constantin Mayer, Sep 24 2015


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.
I. R. Shafarevich, Extensions with prescribed ramification points, Publ. Math., Inst. Hautes Études Sci. 18 (1964), 7195 (Russian). English transl. by J. W. S. Cassels: Am. Math. Soc. Transl., II. Ser., 59 (1966), 128149.  Daniel Constantin Mayer, Sep 24 2015


LINKS



PROG

(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<Cx`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;


CROSSREFS



KEYWORD

hard,more,nonn


AUTHOR



STATUS

approved



