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A227734
Negative fundamental discriminants with noncyclic class groups (negated).
2
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
1,1
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, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
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