A225365 Negative fundamental discriminants with non-isomorphic class groups (negated).

3, 15, 23, 39, 47, 71, 84, 87, 95, 119, 167, 191, 199, 215, 231, 239, 260, 311, 327, 335, 383, 399, 407, 420, 431, 455, 479, 551, 591, 647, 671, 695, 719, 759, 776, 791, 831, 839, 887, 935, 959, 983, 1031, 1079, 1140, 1151, 1199, 1239, 1271, 1295, 1319, 1391, 1439, 1487, 1511, 1559, 1679, 1751, 1799, 1847, 1959, 1991, 2015

Offset: 1

Author

Rick L. Shepherd, May 05 2013

Keywords

• nonn

Comments

More precisely, the least absolute values of negative fundamental discriminants with class groups non-isomorphic to all class groups for negative fundamental discriminants with smaller absolute values.

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 -3 corresponds to the trivial class group. The fundamental discriminant -15 is the first negative fundamental discriminant encountered (least absolute value) whose class group has a different structure, isomorphic to Z_2. The fundamental discriminant -84 is the one with least absolute value whose class group is isomorphic to the Klein-4 group.
		

Links

• Rick L. Shepherd and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 7194 terms from Shepherd)
• 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

Crossrefs

Cf. A003657.

Programs

PARI

{allocatemem(32000000);
\\ Increase precision to more than 100 digits to go beyond 7194 terms.
default(realprecision, 100);
terms_wanted = 7194;
G = Set(); k = 0;
while(length(G)1, print("Certify failed for ", -k, " -- exiting (", length(G), " terms found)"); break);
    if(setsearch(G, F.clgp.cyc)==0,
      G = setunion(G, [F.clgp.cyc]);
      write("b225365.txt", length(G), " ", k);
      write("a225365.txt", length(G), " ", F.clgp.cyc);
      if(length(G)%100==0, print1("...", length(G), "... ")))))}