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
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
KEYWORD
nonn
AUTHOR
Rick L. Shepherd, Jul 28 2013
STATUS
approved