A003657 -id:A003657 - cp's OEIS frontend

A227734 Negative fundamental discriminants with noncyclic class groups (negated).

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

Rick L. Shepherd, Jul 28 2013

nonn

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).

			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.

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
Index entries for sequences related to quadratic fields

Cf. A003657, A003658, A001221, A225365.

{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))))}

A228251 Fundamental discriminant of least absolute value with class group of 2-rank n.

Original entry on oeis.org

-3, 12, 60, -420, 4620, 60060, 1021020, -19399380, 446185740, 12939386460, -401120980260, -14841476269620, -608500527054420, 26165522663340060, -1229779565176982820, -65178316954380089460, 3845520700308425278140, 234576762718813941966540, -15716643102160534111758180

Offset: 0

Rick L. Shepherd, Aug 18 2013

Equivalently, fundamental discriminant of least absolute value with genus group of order 2^n (each such genus group is isomorphic to Z_2 x ... x Z_2 with exactly n copies of Z_2).

The n-th term is the product of n + 1 prime discriminants that are pairwise relatively prime. As the prime discriminants are exactly -8, -4, 8, and +-p for each odd prime p depending upon whether p == 1 (mod 4) or p == 3 (mod 4), respectively, |a(n)| = 2*A002110(n+1) for all n > 1 because usage of -4 precludes usage of +-8 (since the least such product in absolute value is wanted).

			The term a(0) = -3 because -3 is the fundamental discriminant of least absolute value whose corresponding class group, the trivial group, has 2-rank 0 (and its genus group is thus also the trivial group). Being negative, -3 is the discriminant of an imaginary quadratic field.
The term a(2) = 60 (=(-3)(-4)(5)) because its corresponding class group has 2-rank 2 (one fewer than the number of 60's prime discriminant factors); in this case the genus group is isomorphic to Z_2 x Z_2 (as the class group also happens to be here).  As 60 is positive, it is the discriminant of a real quadratic field.

Rick L. Shepherd, Table of n, a(n) for n = 0..348
Index entries for sequences related to quadratic fields

Cf. A003657, A003658, A002110, A100672.

{fd = -3; for(n = 0, 348, if(n > 1, pd = prime(n + 1); if(pd%4 == 3, pd = -pd); fd *= pd, if(n, fd = 12)); write("b228251.txt", n, " ", fd))}

a(n) = ((-1)^k)*2*A002110(n+1) for n > 1, where k is the number of 1 terms from A100672(3) through A100672(n+1) inclusive; a(0) = -3; a(1) = 12.

A306218 Fundamental discriminant D < 0 with the least absolute value such that the first n primes p have (D/p) >= 0, negated.

Original entry on oeis.org

4, 8, 15, 20, 24, 231, 264, 831, 920, 1364, 1364, 9044, 67044, 67044, 67044, 67044, 268719, 268719, 3604695, 4588724, 5053620, 5053620, 5053620, 5053620, 60369855, 364461096, 532735220, 715236599, 1093026360, 2710139064, 2710139064, 3356929784, 3356929784

Offset: 1

Jianing Song, Jan 29 2019

nonn

a(n) is the negated fundamental discriminant D < 0 with the least absolute value such that the first n primes either decompose or ramify in the imaginary quadratic field with discriminant D. See A241482 for the real quadratic field case.

			(-231/2) = 1, (-231/3) = 0, (-231/5) = 1, (-231/7) = 0, (-231/11) = 0, (-231/13) = 1, so 2, 5, 13 decompose in Q[sqrt(-231)] and 3, 7, 11 ramify in Q[sqrt(-231)]. For other fundamental discriminants -231 < D < 0, at least one of 2, 3, 5, 7, 11, 13 is inert in the imaginary quadratic field with discriminant D, so a(6) = 231.

Cf. A003657, A232932, A241482 (the real quadratic field case).

A045535 and A094841 are similar sequences.

a(n) = my(i=1); while(!isfundamental(-i)||sum(j=1, n, kronecker(-i,prime(j))==-1)!=0, i++); i

a(n) = A003657(k), where k is the smallest number such that A232932(k) >= prime(n+1).

a(26)-a(33) from Jinyuan Wang, Apr 06 2019

A106031 a(n) is the number of orbits under the action of GL_2[Z] on the primitive binary quadratic forms of discriminant D, where D < 0 is the n-th fundamental discriminant.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 1, 3, 2, 2, 3, 3, 2, 1, 3, 4, 3, 2, 4, 4, 2, 2, 5, 3, 4, 2, 5, 2, 4, 6, 4, 2, 3, 3, 4, 3, 2, 6, 2, 4, 4, 3, 6, 1, 5, 6, 4, 3, 5, 3, 2, 7

Offset: 1

Steven Finch, May 05 2005

nonn

A006641 is the same except it is under the action of SL_2[Z].

			D = -3, -4, -7, -8, -11, -15, -19, -20, -23, -24, -31, ...,
that is, A003657 negated.

S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Jens Jonasson, Classes of integral binary quadratic forms, Master's thesis (2001), Appendix B.

Cf. A003657, A006641.

cp's OEIS Frontend

A227734 Negative fundamental discriminants with noncyclic class groups (negated).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A228251 Fundamental discriminant of least absolute value with class group of 2-rank n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A306218 Fundamental discriminant D < 0 with the least absolute value such that the first n primes p have (D/p) >= 0, negated.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

Extensions

A106031 a(n) is the number of orbits under the action of GL_2[Z] on the primitive binary quadratic forms of discriminant D, where D < 0 is the n-th fundamental discriminant.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs