A227735 Negative fundamental discriminants with cyclic class groups of composite order (negated).

39, 55, 56, 68, 87, 95, 104, 111, 116, 119, 136, 143, 152, 155, 159, 164, 183, 184, 199, 203, 212, 215, 219, 239, 244, 247, 248, 259, 287, 291, 292, 295, 296, 299, 303, 319, 323, 327, 328, 335, 339, 344, 355, 356, 367, 371, 376, 388, 391, 395, 404, 407, 411

Offset: 1

Author

Rick L. Shepherd, Jul 29 2013

Keywords

• nonn

Comments

Absolute values of fundamental discriminants of imaginary quadratic fields whose class groups are cyclic of composite order. Of course every class group of squarefree order is necessarily cyclic. (This means the negatives of negative fundamental discriminants with class groups of composite squarefree orders are a proper subsequence.)

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

The negative of each term is either a negative fundamental discriminant or the product of exactly one positive prime discriminant and one negative prime discriminant where the product contains at most one factor in {-8, -4, 8} and is unique disregarding order. In the first case, the class number is odd (because the 2-rank of the class group is 0); in the second, even (because the 2-rank is 1).

Examples

			The fundamental discriminant -39 = (-3)(13) has a cyclic class group of order 4, which is composite (but not squarefree). The fundamental discriminant -104 = (-8)(13) has a cyclic class group of order 6, which is composite. The fundamental discriminant -239 is itself a prime discriminant with cyclic class group of order 15, also composite (but not divisible by 2).
		

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, Orders of corresponding class groups
• Index entries for sequences related to quadratic fields

Crossrefs

Cf. A227734.

Programs

PARI

{default(realprecision, 100);
terms_wanted = 10000;
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 &&
      isprime(F.clgp.cyc[1]) == 0,
      t++;
      write("b227735.txt", t, " ", k);
      write("a227735.txt", t, " ", F.clgp.cyc[1]))))}