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A276664 Number of solutions to the congruence y^2 == x^3 - x^2 + 4*x - 4 (mod p) as p runs through the primes. 3
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 (list; graph; refs; listen; history; text; internal format)
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 * A160565 A025252 A177863

Adjacent sequences:  A276661 A276662 A276663 * A276665 A276666 A276667

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Sep 12 2016

STATUS

approved

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Last modified June 25 21:43 EDT 2019. Contains 324357 sequences. (Running on oeis4.)