A276731 Number of solutions to y^2 + y == x^3 - 7 (mod p) as p runs through the primes.

2, 3, 5, 8, 11, 8, 17, 26, 23, 29, 35, 26, 41, 35, 47, 53, 59, 62, 62, 71, 80, 62, 83, 89, 116, 101, 116, 107, 107, 113, 107, 131, 137, 116, 149, 170, 143, 188, 167, 173, 179, 188, 191, 170, 197, 188, 224, 251, 227, 251, 233, 239, 224, 251, 257, 263, 269, 242, 251, 281

Offset

1,1

Comments

This elliptic curve corresponds to a weight 2 newform which is an eta-quotient, namely, (eta(3t)*eta(9t))^2, see Theorem 2 in Martin & Ono.

a(n) is the number of solutions of the congruence y^2 + y == x^3 - 7 (mod prime(n)), n >= 1.

a(n) is also the number of solutions of the congruence y^2 == x^3 - 432 (mod prime(n)), n >= 1.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.

Example

The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used.
The solutions (x, y) of y^2 + y == x^3 - 7 (mod prime(n)) begin:
n, prime(n), a(n)\  solutions (x, y)
1,   2,       2:   (1, 0), (1, 1)
2,   3,       3:   (0, 1), (1, 0), (1, 2)
3,   5,       5:   (2, 2), (3, 0), (3, 4),
                   (4, 1), (4, 3)
4,   7,       8:   (0, 0), (0, 6), (3, 2),
                   (3, 4), (5, 2), (5, 4),
                   (6, 2), (6, 4)

Prog

(Ruby)
require 'prime'
def A(a3, a2, a4, a6, n)
  ary = []
  Prime.take(n).each{|p|
    a = Array.new(p, 0)
    (0..p - 1).each{|i| a[(i * i + a3 * i) % p] += 1}
    ary << (0..p - 1).inject(0){|s, i| s + a[(i * i * i + a2 * i * i + a4 * i + a6) % p]}
  }
  ary
end
def A276731(n)
  A(1, 0, 0, -7, n)
end

Crossrefs

Sequence in context: A023176 A063178 A280205 * A028891 A028890 A189716

Adjacent sequences: A276728 A276729 A276730 * A276732 A276733 A276734

Keyword

nonn

Author

Seiichi Manyama, Sep 16 2016

Status

approved