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A227734 Negative fundamental discriminants with noncyclic class groups (negated).

%I A227734 #18 Dec 26 2015 02:21:47
%S A227734 84,120,132,168,195,228,231,255,260,264,276,280,308,312,340,372,399,
%T A227734 408,420,435,440,455,456,483,516,520,532,552,555,564,580,595,615,616,
%U A227734 627,644,651,660,663,680,696,708,715,728,740,744,759,760,795,804,820,836,840
%N A227734 Negative fundamental discriminants with noncyclic class groups (negated).
%C A227734 Absolute values of discriminants of imaginary quadratic fields whose class groups are noncyclic.
%C A227734 The n-th line of the linked file gives the invariant factor decomposition of the class group corresponding to the fundamental discriminant -a(n).
%H A227734 Rick L. Shepherd, <a href="/A227734/b227734.txt">Table of n, a(n) for n = 1..10000</a>
%H A227734 Rick L. Shepherd, <a href="http://libres.uncg.edu/ir/uncg/f/Shepherd_uncg_0154M_11099.pdf">Binary quadratic forms and genus theory</a>, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
%H A227734 Rick L. Shepherd, <a href="/A227734/a227734_1.txt">Invariant factor decompositions for corresponding class groups</a>
%H A227734 <a href="/index/Qua#quadfield">Index entries for sequences related to quadratic fields</a>
%e A227734 The fundamental discriminant -231 = (-3)(-7)(-11) has class group isomorphic to Z_6 x Z_2. The fundamental discriminant -420 = (-7)(-4)(-3)(5) has class group isomorphic to Z_2 x Z_2 x Z_2. The fundamental discriminant (also prime discriminant) -3299 has class group isomorphic to Z_9 x Z_3. The fundamental discriminant -3896 = 8(-147) has class group isomorphic to Z_12 x Z_3. Here and in general for fundamental discriminants, the 2-rank of each class group is the number of prime discriminant factors minus one.
%o A227734 (PARI)
%o A227734 {default(realprecision, 100);
%o A227734 terms_wanted = 100000;
%o A227734 t = 0; k = 0;
%o A227734 while(t < terms_wanted,
%o A227734   k++;
%o A227734   if(isfundamental(-k),
%o A227734     F = bnfinit(quadpoly(-k, x), , [6, 6, 4]);
%o A227734     if(bnfcertify(F) <> 1,
%o A227734       print("Certify failed for ", -k, " -- exiting (",
%o A227734         t, " terms found)"); break);
%o A227734     if(length(F.clgp.cyc) > 1,
%o A227734       t++;
%o A227734       write("b227734.txt", t, " ", k);
%o A227734       write("a227734.txt", t, " ", F.clgp.cyc))))}
%Y A227734 Cf. A003657, A003658, A001221, A225365.
%K A227734 nonn
%O A227734 1,1
%A A227734 _Rick L. Shepherd_, Jul 28 2013