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

Author

Rick L. Shepherd, Jul 28 2013

Keywords

• nonn

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).

Examples

			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.
		

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

Crossrefs

Cf. A003657, A003658, A001221, A225365.

Programs

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))))}