%I #18 Dec 26 2015 02:21:47

%S 84,120,132,168,195,228,231,255,260,264,276,280,308,312,340,372,399,

%T 408,420,435,440,455,456,483,516,520,532,552,555,564,580,595,615,616,

%U 627,644,651,660,663,680,696,708,715,728,740,744,759,760,795,804,820,836,840

%N Negative fundamental discriminants with noncyclic class groups (negated).

%C Absolute values of discriminants of imaginary quadratic fields whose class groups are noncyclic.

%C 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 Rick L. Shepherd, <a href="/A227734/b227734.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 Rick L. Shepherd, <a href="/A227734/a227734_1.txt">Invariant factor decompositions for corresponding class groups</a>

%H <a href="/index/Qua#quadfield">Index entries for sequences related to quadratic fields</a>

%e 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 (PARI)

%o {default(realprecision, 100);

%o terms_wanted = 100000;

%o t = 0; k = 0;

%o while(t < terms_wanted,

%o k++;

%o if(isfundamental(-k),

%o F = bnfinit(quadpoly(-k, x), , [6, 6, 4]);

%o if(bnfcertify(F) <> 1,

%o print("Certify failed for ", -k, " -- exiting (",

%o t, " terms found)"); break);

%o if(length(F.clgp.cyc) > 1,

%o t++;

%o write("b227734.txt", t, " ", k);

%o write("a227734.txt", t, " ", F.clgp.cyc))))}

%Y Cf. A003657, A003658, A001221, A225365.

%K nonn

%O 1,1

%A _Rick L. Shepherd_, Jul 28 2013