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A276807 Number of solutions of the congruence y^2 == x^3 - x^2 - 4*x + 4 (mod p) as p runs through the primes.

%I A276807 #19 Oct 24 2018 02:32:39
%S A276807 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 A276807 95,119,87,119,111,95,135,135,143,151,135,167,159,151,143,167,167,175,
%U A276807 191,191,215,183,231,231,215,207,223,255,223,231,255,271,279,263,303,255
%N A276807 Number of solutions of the congruence y^2  == x^3 - x^2 - 4*x + 4 (mod p) as p runs through the primes.
%C A276807 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 A276807 Seiichi Manyama, <a href="/A276807/b276807.txt">Table of n, a(n) for n = 1..10000</a>
%H A276807 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 A276807 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 A276807 The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used.
%e A276807 The solutions (x, y) of y^2 == x^3 - x^2 - 4*x + 4 (mod prime(n)) begin:
%e A276807 n, prime(n), a(n)  solutions (x, y)
%e A276807 1,   2,       2:   (0, 0), (1, 0)
%e A276807 2,   3,       4:   (0, 1), (0, 2), (1, 0),
%e A276807                    (2, 0)
%e A276807 3,   5,       7:   (0, 2), (0, 3), (1, 0),
%e A276807                    (2, 0), (3, 0), (4, 1),
%e A276807                    (4, 4)
%e A276807 4,   7,       7:   (0, 2), (0, 5), (1, 0),
%e A276807                    (2, 0), (4, 1), (4, 6),
%e A276807                    (5, 0)
%o A276807 (Ruby)
%o A276807 require 'prime'
%o A276807 def A(a3, a2, a4, a6, n)
%o A276807   ary = []
%o A276807   Prime.take(n).each{|p|
%o A276807     a = Array.new(p, 0)
%o A276807     (0..p - 1).each{|i| a[(i * i + a3 * i) % p] += 1}
%o A276807     ary << (0..p - 1).inject(0){|s, i| s + a[(i * i * i + a2 * i * i + a4 * i + a6) % p]}
%o A276807   }
%o A276807   ary
%o A276807 end
%o A276807 def A276807(n)
%o A276807   A(0, -1, -4, 4, n)
%o A276807 end
%Y A276807 Cf. A276649.
%K A276807 nonn
%O A276807 1,1
%A A276807 _Seiichi Manyama_, Sep 17 2016