A276807 Number of solutions of the congruence y^2 == x^3 - x^2 - 4*x + 4 (mod p) as p runs through the primes.

2, 4, 7, 7, 7, 15, 15, 23, 31, 23, 23, 31, 47, 39, 47, 55, 55, 63, 71, 63, 63, 87, 87, 95, 95, 119, 87, 119, 111, 95, 135, 135, 143, 151, 135, 167, 159, 151, 143, 167, 167, 175, 191, 191, 215, 183, 231, 231, 215, 207, 223, 255, 223, 231, 255, 271, 279, 263, 303, 255

Offset: 1

Author

Seiichi Manyama, Sep 17 2016

Keywords

• nonn

Comments

This elliptic curve corresponds to a weight 2 newform which is an eta-quotient, namely, eta(2t)*eta(4t)*eta(6t)*eta(12t), see Theorem 2 in Martin & Ono.

Examples

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

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.

Crossrefs

Cf. A276649.

Programs

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 A276807(n)
  A(0, -1, -4, 4, n)
end

Formula

a(n) gives the number of solutions of the congruence y^2 == x^3 - x^2 - 4*x + 4 (mod prime(n)), n >= 1.