A275742 Number of solutions to the congruence y^2 + x*y + y == x^3 + x^2 - 10*x - 10 (mod p) as p runs through the primes.

3, 4, 4, 7, 15, 15, 15, 15, 23, 31, 31, 47, 31, 39, 39, 63, 63, 63, 55, 79, 63, 79, 71, 95, 95, 95, 119, 119, 95, 111, 135, 143, 143, 143, 127, 159, 143, 167, 167, 191, 159, 191, 175, 191, 191, 207, 191, 215, 247, 223, 239, 255, 255, 239, 239, 247, 255, 255, 271, 287

Offset

1,1

Comments

This elliptic curve corresponds to a weight 2 newform which is an eta-quotient, namely, eta(t)*eta(3t)*eta(5t)*eta(15t), see Theorem 2 in Martin & Ono. - Charles R Greathouse IV, Sep 14 2016

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 + x^2 - 10*x - 10 (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 + x^2 - 10*x - 10 (mod prime(n)) begin:
n, prime(n), a(n)  solutions (x, y)
1,   2,       3:   (0, 0), (0, 1), (1, 0)
2,   3,       4:   (0, 1), (1, 0), (1, 1),
                   (2, 0)
3,   5,       4:   (0, 0), (0, 4), (3, 3),
                   (4, 0)
4,   7,       7:   (1, 1), (1, 4), (2, 2),
                   (3, 5), (5, 3), (5, 5),
                   (6, 0)

Prog

(PARI) a(n, p=prime(n))=sum(x=1, p, sum(y=1, p, (y^2+x*y+y-x^3-x^2+10*x+10)%p==0)) \\ Charles R Greathouse IV, Sep 12 2016
(PARI) a(n, p=prime(n))=my(y='y); sum(x=1, p, #polrootsmod(y^2+x*y+y-x^3-x^2+10*x+10, p)) \\ Charles R Greathouse IV, Sep 12 2016

Crossrefs

Cf. A275745.

Sequence in context: A342332 A225738 A186679 * A161496 A010612 A021033

Adjacent sequences: A275739 A275740 A275741 * A275743 A275744 A275745

Keyword

nonn

Author

Seiichi Manyama, Sep 10 2016

Extensions

Terms corrected by Charles R Greathouse IV, Sep 12 2016

Status

approved