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%I #12 Sep 08 2022 08:46:10
%S 229,257,316,321,473,568,697,761,785,892,940,985,993,1016,1229,1304,
%T 1345,1384,1436,1509,1765,1929,2024,2089,2101,2233,2296,2505,2920,2993
%N Fundamental discriminants d uniquely characterizing all complex biquadratic fields Q(sqrt(-3),sqrt(d)) which have 3-class group of type (3,3) and abelian 3-class field tower of length 1.
%C This is the beginning of an investigation of the maximal unramified pro-3 extension of complex bicyclic biquadratic fields containing the third roots of unity which have an elementary 3-class group of rank two.
%C For the discriminants d in A250237, the 3-class field tower of K=Q(sqrt(-3),sqrt(d)) is abelian, terminating with the first stage at the Hilbert 3-class field already. An equivalent condition is that the second 3-class group G of K is given by G=SmallGroup(9,2). Another equivalent condition in terms of a fundamental system of units has been given by Yoshida.
%D 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.
%H D. C. Mayer, <a href="http://www.worldscientific.com/doi/abs/10.1142/S179304211250025X">The second p-class group of a number field</a>, Int. J. Number Theory 8 (2) (2012), 471-505.
%H E. Yoshida, <a href="http://dx.doi.org/10.4064/aa107-4-2">On the 3-class field tower of some biquadratic fields</a>, Acta Arith. 107 (2003), no. 4, 327-336.
%e A250237 covers the dominant part of A250236. The smallest discriminant d in A250236 with non-abelian 3-class field tower of length bigger than 1 is given by d=A250238(1)=469, the initial term of the disjoint sequence A250238.
%o (Magma)SetClassGroupBounds("GRH"); for n := 229 to 3000 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 (0 eq #pPrimaryInvariants(CH, 3)) then n, ", "; end if; end if; end if; end for;
%Y A006832, A250235, A250236 are supersequences, A250238, A250239, A250240, A250241, A250242 are disjoint sequences.
%K hard,nonn
%O 1,1
%A _Daniel Constantin Mayer_, Nov 15 2014
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