login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A225365 Negative fundamental discriminants with non-isomorphic 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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).

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 Klein-4 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 15 22:02 EST 2019. Contains 330012 sequences. (Running on oeis4.)