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 A250237 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. 7
 229, 257, 316, 321, 473, 568, 697, 761, 785, 892, 940, 985, 993, 1016, 1229, 1304, 1345, 1384, 1436, 1509, 1765, 1929, 2024, 2089, 2101, 2233, 2296, 2505, 2920, 2993 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. 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. 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. LINKS D. C. Mayer, The second p-class group of a number field, Int. J. Number Theory 8 (2) (2012), 471-505. E. Yoshida, On the 3-class field tower of some biquadratic fields, Acta Arith. 107 (2003), no. 4, 327-336. EXAMPLE 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. PROG (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 : 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; CROSSREFS A006832, A250235, A250236 are supersequences, A250238, A250239, A250240, A250241, A250242 are disjoint sequences. Sequence in context: A250235 A250236 A094612 * A112847 A157348 A142221 Adjacent sequences:  A250234 A250235 A250236 * A250238 A250239 A250240 KEYWORD hard,nonn AUTHOR Daniel Constantin Mayer, Nov 15 2014 STATUS approved

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Last modified November 19 08:40 EST 2018. Contains 317347 sequences. (Running on oeis4.)