

A225365


Negative fundamental discriminants with nonisomorphic class groups (negated).


2



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
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OFFSET

1,1


COMMENTS

More precisely, the least absolute values of negative fundamental discriminants with class groups nonisomorphic to all class groups for negative fundamental discriminants with smaller absolute values.
The nth 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 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


EXAMPLE

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 Klein4 group.


PROG

(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)<terms_wanted,
k++;
if(isfundamental(k),
F = bnfinit(quadpoly(k, x), , [6, 6, 4]); \\ Without optional 3rd argument, Generalized Riemann Hypothesis assumed
if(bnfcertify(F)<>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), "... ")))))}


CROSSREFS

Cf. A003657.
Sequence in context: A083934 A072200 A106403 * A225060 A060649 A009210
Adjacent sequences: A225362 A225363 A225364 * A225366 A225367 A225368


KEYWORD

nonn


AUTHOR

Rick L. Shepherd, May 05 2013


STATUS

approved



