

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
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OFFSET

1,1


COMMENTS

This elliptic curve corresponds to a weight 2 newform which is an etaquotient, namely, eta(t)*eta(3t)*eta(5t)*eta(15t), see Theorem 2 in Martin & Ono.  Charles R Greathouse IV, Sep 14 2016


LINKS



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+yx^3x^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+yx^3x^2+10*x+10, p)) \\ Charles R Greathouse IV, Sep 12 2016


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



