A319659 2-rank of the class group of imaginary quadratic field with discriminant -k, k = A003657(n).

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 2, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 0, 0, 2, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 0, 1, 1, 1, 0, 0, 2, 2, 1, 1, 0, 1, 1, 1, 0, 2, 1, 2, 0, 2, 1, 0, 2, 2

Offset

1,27

Comments

The p-rank of a finite abelian group G is equal to log_p(#{x belongs to G : x^p = 1}) where p is a prime number. In this case, G is the class group of Q(sqrt(-k)), and #{x belongs to G : x^p = 1} is the number of genera of Q(sqrt(-k)) (cf. A003640).

From Jianing Song, Jun 10 2026: (Start)

Also the 2-rank of the narrow class group, and the 2-rank of the Pólya group of imaginary quadratic field with discriminant -A003657(n). The Pólya group of a number field K is the subgroup of class group Cl(K) generated by {products of prime ideals of O_K with norm q : q prime powers}.

For imaginary quadratic fields the three 2-ranks are the same, but for real quadratic fields they are different: see A317991, A391436, and A396865. (End)

Links

Table of n, a(n) for n=1..87.

Jean-Luc Chabert, From Pólya fields to Pólya groups (I) Galois extensions, Journal of Number Theory, 2019, 203, pp.360-375.

Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.

Formula

a(n) = log_2(A003640(n)) = omega(A003657(n)) - 1, where omega(k) is the number of distinct prime divisors of k.

Mathematica

PrimeNu[Select[Range[300], MemberQ[{3, 4, 7, 8, 11, 15}, Mod[#, 16]] && SquareFreeQ[#/2^IntegerExponent[#, 2]] &]] - 1 (* Amiram Eldar, Mar 28 2026 *)

Prog

(PARI) for(n=1, 1000, if(isfundamental(-n), print1(omega(n) - 1, ", ")))

Crossrefs

Cf. A003640, A003657, A319660, A319661, A319662, A396867 (earliest occurrences of each number).

Cf. A317991, A391436, A396865.

Sequence in context: A116663 A258940 A340607 * A050372 A037802 A037879

Adjacent sequences: A319656 A319657 A319658 * A319660 A319661 A319662

Keyword

nonn

Author

Jianing Song, Sep 25 2018

Status

approved