The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 4
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Sep 10 2016
EXTENSIONS
Terms corrected by Charles R Greathouse IV, Sep 12 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 | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 29 20:04 EDT 2024. Contains 372952 sequences. (Running on oeis4.)