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 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 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 Cf. A275745. Sequence in context: A342332 A225738 A186679 * A161496 A010612 A021033 Adjacent sequences: A275739 A275740 A275741 * A275743 A275744 A275745 KEYWORD nonn AUTHOR Seiichi Manyama, Sep 10 2016 EXTENSIONS Terms corrected by Charles R Greathouse IV, Sep 12 2016 STATUS approved

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