login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A272197 Number of solutions of the congruence y^2 == x^3 + 1 (mod p) as p runs through the primes. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

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.

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:  (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)

CROSSREFS

Cf. A000040, A000727, A272198.

Sequence in context: A233098 A258975 A111214 * A207982 A259155 A210144

Adjacent sequences:  A272194 A272195 A272196 * A272198 A272199 A272200

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, May 02 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 15 01:50 EDT 2021. Contains 343909 sequences. (Running on oeis4.)