A276664 Number of solutions to the congruence y^2 == x^3 - x^2 + 4*x - 4 (mod p) as p runs through the primes.

2, 1, 6, 9, 11, 11, 23, 15, 29, 23, 27, 35, 35, 33, 41, 59, 71, 59, 69, 59, 71, 87, 89, 95, 95, 95, 117, 101, 107, 119, 129, 131, 119, 135, 155, 171, 179, 153, 185, 179, 167, 191, 179, 167, 179, 207, 195, 213, 221, 215, 239, 215, 227, 251, 263, 245, 251, 291, 251

Offset

1,1

Comments

This elliptic curve corresponds to a weight 2 newform which is an eta-quotient, namely, (eta(4t)*eta(20t))^6 / (eta(2t)*eta(8t)*eta(10t)*eta(40t))^2, 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,       1:   (1, 0)
3,   5,       6:   (0, 1), (0, 4), (1, 0),
                   (3, 1), (3, 4), (4, 0)
4,   7,       9:   (1, 0), (2, 1), (2, 6),
                   (4, 2), (4, 5), (5, 2),
                   (5, 5), (6, 2), (6, 5)

Prog

(Ruby)
require 'prime'
def A276664(n)
  ary = []
  Prime.take(n).each{|p|
    a = Array.new(p, 0)
    (0..p - 1).each{|i| a[(i * i) % p] += 1}
    ary << (0..p - 1).inject(0){|s, i| s + a[(i * i * i - i * i + 4 * i - 4) % p]}
  }
  ary
end

Crossrefs

Sequence in context: A176013 A263255 A145663 * A335663 A160565 A025252

Adjacent sequences: A276661 A276662 A276663 * A276665 A276666 A276667

Keyword

nonn

Author

Seiichi Manyama, Sep 12 2016

Status

approved