A276173 Number of solutions to the congruence y^2 + x*y + y == x^3 + 4*x - 6 (mod p) as p runs through the primes.

3, 5, 5, 6, 11, 17, 11, 17, 23, 35, 35, 35, 35, 35, 59, 47, 65, 53, 71, 71, 71, 71, 89, 95, 107, 101, 107, 95, 107, 107, 143, 113, 119, 125, 167, 143, 161, 179, 179, 185, 191, 161, 167, 179, 215, 179, 215, 215, 209, 233, 239, 215, 251, 269, 239, 263, 281, 287, 287, 287

Offset

1,1

Comments

This elliptic curve corresponds to a weight 2 newform which is an eta-quotient, namely, eta(t)*eta(2t)*eta(7t)*eta(14t), see Theorem 2 in Martin & Ono.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..1000

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*y + y == x^3 + 4*x - 6 (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*y + y == x^3 + 4*x - 6 (mod prime(n)) begin:
n, prime(n), a(n)\  solutions (x, y)
1,   2,       3:   (0, 0), (0, 1), (1, 1)
2,   3,       5:   (0, 0), (0, 2), (1, 2),
                   (2, 1), (2, 2)
3,   5,       5:   (1, 4), (2, 0), (2, 2),
                   (4, 2), (4, 3)
4,   7,       6:   (1, 6), (2, 2), (3, 1),
                   (3, 2), (5, 3), (5, 5)

Crossrefs

Cf. A030187, A275742, A276174.

Sequence in context: A010616 A296485 A055594 * A197631 A266567 A282624

Adjacent sequences: A276170 A276171 A276172 * A276174 A276175 A276176

Keyword

nonn

Author

Seiichi Manyama, Sep 10 2016

Status

approved