

A250241


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(729,34).


7



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

1,1


COMMENTS

For the discriminants d in A250241, 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(729,34), which is called the nonCF group H by Ascione, Havas and LeedhamGreen. It has properties very similar to those of SmallGroup(729,37), called the nonCF 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 3class group of the Hilbert 3class 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 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

Table of n, a(n) for n=1..25.
J. A. Ascione, G. Havas, and C. R. LeedhamGreen, A computer aided classification of certain groups of prime power order, Bull. Austral. Math. Soc. 17 (1977), 257274.
D. C. Mayer, The second pclass group of a number field, Int. J. Number Theory 8 (2) (2012), 471505.
D. C. Mayer, The second pclass group of a number field. Preprint: arXiv:1403.3899v1 [math.NT], 2014.
D. C. Mayer, Principalization algorithm via class group structure, Preprint: arXiv:1403.3839v1 [math.NT], 2014. J. Théor. Nombres Bordeaux 26 (2014), no. 2, 415464.


PROG

(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<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, 3, 3] eq pPrimaryInvariants(CH, 3)) then n, ", "; end if; end if; end if; end for;


CROSSREFS

A006832, A250235, A250236 are supersequences.
A250237, A250238, A250239, A250240, A250242 are disjoint sequences.
Sequence in context: A260840 A252310 A236351 * A252152 A035894 A179702
Adjacent sequences: A250238 A250239 A250240 * A250242 A250243 A250244


KEYWORD

hard,more,nonn


AUTHOR

Daniel Constantin Mayer, Nov 15 2014


STATUS

approved



