A276807 - OEIS

%I #19 Oct 24 2018 02:32:39

%S 2,4,7,7,7,15,15,23,31,23,23,31,47,39,47,55,55,63,71,63,63,87,87,95,

%T 95,119,87,119,111,95,135,135,143,151,135,167,159,151,143,167,167,175,

%U 191,191,215,183,231,231,215,207,223,255,223,231,255,271,279,263,303,255

%N Number of solutions of the congruence y^2  == x^3 - x^2 - 4*x + 4 (mod p) as p runs through the primes.

%C This elliptic curve corresponds to a weight 2 newform which is an eta-quotient, namely, eta(2t)*eta(4t)*eta(6t)*eta(12t), see Theorem 2 in Martin & Ono.

%H Seiichi Manyama, <a href="/A276807/b276807.txt">Table of n, a(n) for n = 1..10000</a>

%H Yves Martin and Ken Ono, <a href="http://dx.doi.org/10.1090/S0002-9939-97-03928-2">Eta-Quotients and Elliptic Curves</a>, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.

%F a(n) gives the number of solutions of the congruence y^2 == x^3 - x^2 - 4*x + 4 (mod prime(n)), n >= 1.

%e The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used.

%e The solutions (x, y) of y^2 == x^3 - x^2 - 4*x + 4 (mod prime(n)) begin:

%e n, prime(n), a(n)  solutions (x, y)

%e 1,   2,       2:   (0, 0), (1, 0)

%e 2,   3,       4:   (0, 1), (0, 2), (1, 0),

%e                    (2, 0)

%e 3,   5,       7:   (0, 2), (0, 3), (1, 0),

%e                    (2, 0), (3, 0), (4, 1),

%e                    (4, 4)

%e 4,   7,       7:   (0, 2), (0, 5), (1, 0),

%e                    (2, 0), (4, 1), (4, 6),

%e                    (5, 0)

%o (Ruby)

%o require 'prime'

%o def A(a3, a2, a4, a6, n)

%o   ary = []

%o   Prime.take(n).each{|p|

%o     a = Array.new(p, 0)

%o     (0..p - 1).each{|i| a[(i * i + a3 * i) % p] += 1}

%o     ary << (0..p - 1).inject(0){|s, i| s + a[(i * i * i + a2 * i * i + a4 * i + a6) % p]}

%o   }

%o   ary

%o end

%o def A276807(n)

%o   A(0, -1, -4, 4, n)

%o end

%Y Cf. A276649.

%K nonn

%O 1,1

%A _Seiichi Manyama_, Sep 17 2016