A272207 Number of solutions to the congruence y^2 == x^3 + x^2 + 4*x + 4 (mod p) as p runs through the primes.

2, 5, 6, 5, 11, 11, 23, 23, 17, 23, 35, 35, 35, 53, 53, 59, 47, 59, 65, 83, 71, 71, 77, 95, 95, 95, 89, 113, 107, 119, 125, 131, 119, 143, 155, 131, 179, 173, 149, 179, 191, 191, 203, 167, 179, 191, 227, 233, 233, 215, 239, 263, 227, 251, 263, 281, 251, 251, 251, 275

Offset: 1

Author

Wolfdieter Lang, May 20 2016

Keywords

• nonn
• easy

Comments

In the Martin and Ono reference, in Theorem 2, this elliptic curve appears in the fourth row, starting with conductor 20, as a strong Weil curve for the weight 2 newform (eta(2*z)*eta(10*z))^2, symbolically 2^2 10^2, with Im(z) > 0, and the Dedekind eta function. See A030205 which gives the q-expansion (q = exp(2*Pi*i*z)) of exp(-Pi*i*z)*(eta(z)*eta(5*z))^2. For the q-expansion of (eta(2*z)*eta(10*z))^2 one has interspersed 0's: 0, 1, 0, -2, 0, -1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 2, 0, -6, ... This modular cusp form of weight 2 appears as the 39th entry in Martin's Table I.

For the p-defect prime(n) - a(n) see A273163(n), n >= 1.

The discriminant of this elliptic curve is -400 = -2^4*5^2 (bad primes 2 and 5, also the prime divisors of the conductor).

The congruence y^2 == x^3 + x^2 - x has the same number of solutions modulo prime(n). See a comment on A030205. The discriminant equals +5.

Examples

			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,       5:   (0, 1), (0, 2), (1, 1),
                   (1, 2) (2, 0)
3,   5,       6:   (0, 2), (0, 3), (1, 0),
                   (2, 2), (2, 3), (4, 0)
4,   7,       5:   (0, 2), (0, 5), (4, 3),
                   (4, 4), (6, 0)
5,  11,      11:   (0, 2), (0, 9), (4, 1),
                   (4, 10), (5, 3), (5, 8),
                   (6, 4), (6, 7), (9, 5),
                   (9, 6), (10, 0)
...
The solutions (x, y) of y^2 == x^3 + x^2 - x (mod prime(n)) begin:
n, prime(n), a(n)\  solutions (x, y)
1,   2,       2:   (0, 0), (1, 1)
2,   3,       5:   (0, 0), (1, 1), (1, 2),
                   (2, 1) (2, 2)
3,   5,       6:   (0, 0), (1, 1), (1, 4),
                   (2, 0), (4, 1), (4, 4)
4,   7,       5:   (0, 0), (1, 1), (1, 6),
                   (6, 1), (6, 6)
5,  11,      11:   (0, 0), (1, 1), (1, 10),
                   (3, 0), (6, 2), (6, 9),
                   (7, 0), (9, 3), (9, 8),
                   (10, 1), (10, 10)
...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
• Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.

Crossrefs

Cf. A000040, A030205, A273163.

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.

a(n) gives also the number of solutions of the congruence y^2 == x^3 + x^2 - x (mod prime(n)), n >= 1.