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
Seiichi Manyama, Table of n, a(n) for n = 1..1000
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
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Sep 10 2016
EXTENSIONS
Terms corrected by Charles R Greathouse IV, Sep 12 2016
STATUS
approved