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A272202 Number of solutions of the congruence y^2 == x^3 - 1 (mod p) as p runs through the primes. 2
2, 3, 5, 3, 11, 11, 17, 27, 23, 29, 27, 47, 41, 51, 47, 53, 59, 47, 51, 71, 83, 75, 83, 89, 83, 101, 123, 107, 107, 113, 147, 131, 137, 123, 149, 147, 143, 171, 167, 173, 179, 155, 191, 191, 197, 171, 195, 195, 227, 251, 233, 239, 227, 251, 257, 263, 269, 243, 251, 281 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

In the Martin and Ono reference, in Theorem 2, this elliptic curve appears in the last column, starting with Conductor 144, as a strong Weil curve for the weight 2 newform  eta^{12}(12*z) / (eta^4(6*z) * eta^4(24*z)), symbolically 12^{12} 6^{-4} 24^{-4}, with Im(z) > 0, and the Dedekind eta function. See A187076 which gives the q-expansion (q = exp(2*Pi*i*z)) of exp(-Pi*i*z/3)* eta(2*z)^{12} / (eta^4(z)*eta^4(4*z)). For the q-expansion of 12^{12} 6^{-4} 24^{-4} one has a leading zero and 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.

For the elliptic curve y^2 == x^3 + 1 (mod prime(n)) see A000727, A272197, A272198, A272200 and A272201.

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 - 1 (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 - 1 (mod prime(n)) begin:

n, prime(n), a(n)\ solutions (x, y)

1,    2,      2:   (0, 1), (1, 0)

2,    3,      3:   (1, 0), (2, 1), (2, 2)

3,    5,      5:   (0, 2), (0, 3), (1, 0),

                   (3, 1), (3, 4)

4,    7,      3:   (1, 0), (2, 0), (4, 0)

5,    11,    11:   (1, 0), (3, 2), (3, 9),

                   (5, 5), (5, 6), (7, 1),

                   (7, 10), (8, 4), (8, 7),

                   (10, 3), (10, 8)

...

CROSSREFS

Cf. A000040, A187076, A272203.

Sequence in context: A105574 A105562 A323704 * A244609 A209195 A113222

Adjacent sequences:  A272199 A272200 A272201 * A272203 A272204 A272205

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, May 05 2016

STATUS

approved

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Last modified April 23 01:31 EDT 2021. Contains 343198 sequences. (Running on oeis4.)