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

1,1

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.

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.

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.

Example

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)

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

Crossrefs

Cf. A276649.

Sequence in context: A154614 A342616 A161211 * A363166 A161368 A023978

Adjacent sequences: A276804 A276805 A276806 * A276808 A276809 A276810

Keyword

nonn

Author

Seiichi Manyama, Sep 17 2016

Status

approved