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

84, 120, 132, 168, 195, 228, 231, 255, 260, 264, 276, 280, 308, 312, 340, 372, 399, 408, 420, 435, 440, 455, 456, 483, 516, 520, 532, 552, 555, 564, 580, 595, 615, 616, 627, 644, 651, 660, 663, 680, 696, 708, 715, 728, 740, 744, 759, 760, 795, 804, 820, 836, 840

Offset

1,1

Comments

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

The n-th line of the linked file gives the invariant factor decomposition of the class group corresponding to the fundamental discriminant -a(n).

Links

Rick L. Shepherd, Table of n, a(n) for n = 1..10000

Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.

Rick L. Shepherd, Invariant factor decompositions for corresponding class groups

Index entries for sequences related to quadratic fields

Example

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.

Prog

(PARI)
{default(realprecision, 100);
terms_wanted = 100000;
t = 0; k = 0;
while(t < terms_wanted,
  k++;
  if(isfundamental(-k),
    F = bnfinit(quadpoly(-k, x), , [6, 6, 4]);
    if(bnfcertify(F) <> 1,
      print("Certify failed for ", -k, " -- exiting (",
        t, " terms found)"); break);
    if(length(F.clgp.cyc) > 1,
      t++;
      write("b227734.txt", t, " ", k);
      write("a227734.txt", t, " ", F.clgp.cyc))))}

Crossrefs

Cf. A003657, A003658, A001221, A225365.

Sequence in context: A219801 A316833 A316834 * A192322 A015708 A114822

Adjacent sequences: A227731 A227732 A227733 * A227735 A227736 A227737

Keyword

nonn

Author

Rick L. Shepherd, Jul 28 2013

Status

approved