A272197 Number of solutions of the congruence y^2 == x^3 + 1 (mod p) as p runs through the primes.

2, 3, 5, 11, 11, 11, 17, 11, 23, 29, 35, 47, 41, 35, 47, 53, 59, 47, 83, 71, 83, 83, 83, 89, 83, 101, 83, 107, 107, 113, 107, 131, 137, 155, 149, 155, 143, 155, 167, 173, 179, 155, 191, 191, 197, 227, 227, 251, 227, 251, 233, 239, 227, 251, 257, 263, 269, 299, 251, 281

Offset: 1

Author

Wolfdieter Lang, May 02 2016

Keywords

• nonn
• easy

Comments

This elliptic curve is discussed in the Silverman reference. In the table the p-defects prime(n) - a(n) are shown for primes 2 to 113.

In the Martin and Ono reference, in Theorem 2, this elliptic curve appears in the eighth row, starting with Conductor 36, as a strong Weil curve for the weight 2 newform eta(6*z)^4, with Im(z) > 0, and the Dedekind eta function. See A000727 which gives the q-expansion (q = exp(2*Pi*i*z)) of exp(-Pi*i*z/3)*eta(z)^4. For the q-expansion of eta(6*z)^4 one has 5 interspersed 0's: 0,1,0,0,0,0,0,-4,0,0,0,0,0,2,0,0,0,0,0,8,...

The discriminant of this elliptic curve is -3^3 = -27.

Examples

			The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used. The solutions (x, y) of y^2  == x^3 + 1 (mod prime(n)) begin:
n, prime(n), a(n)\  solutions (x, y)
1,   2,       2:  (0, 1), (1, 0)
2,   3,       3:  (0, 1), (0, 2), (2, 0)
3,   5,       5:  (0, 1), (0, 4), (2, 2),
                  (2, 3), (4, 0)
4,   7,      11:  (0, 1), (0, 6), (1, 3),
                  (1, 4), (2, 3), (2, 4),
                  (3, 0), (4, 3), (4, 4),
                  (5, 0), (6, 0)
5,  11,      11:  (0, 1), (0, 10), (2, 3),
                  (2, 8), (5, 4), (5, 7),
                  (7, 5), (7, 6), (9, 2),
                  (9, 9), (10, 0)
		

References

• J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, Exercise 45.5, p. 405, Exercise 47.2, p. 415. (4th ed., Pearson 2014, Exercise 5, p. 371, Exercise 2, p. 385).

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.

Crossrefs

Cf. A000040, A000727, A272198.

Formula

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