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A276664 Number of solutions to the congruence y^2 == x^3 - x^2 + 4*x - 4 (mod p) as p runs through the primes.

%I A276664 #21 Oct 24 2018 08:07:57
%S A276664 2,1,6,9,11,11,23,15,29,23,27,35,35,33,41,59,71,59,69,59,71,87,89,95,
%T A276664 95,95,117,101,107,119,129,131,119,135,155,171,179,153,185,179,167,
%U A276664 191,179,167,179,207,195,213,221,215,239,215,227,251,263,245,251,291,251
%N A276664 Number of solutions to the congruence y^2 == x^3 - x^2 + 4*x - 4 (mod p) as p runs through the primes.
%C A276664 This elliptic curve corresponds to a weight 2 newform which is an eta-quotient, namely, (eta(4t)*eta(20t))^6 / (eta(2t)*eta(8t)*eta(10t)*eta(40t))^2, see Theorem 2 in Martin & Ono.
%H A276664 Seiichi Manyama, <a href="/A276664/b276664.txt">Table of n, a(n) for n = 1..10000</a>
%H A276664 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 A276664 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 A276664 The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used.
%e A276664 The solutions (x, y) of y^2 == x^3 - x^2 + 4*x - 4 (mod prime(n)) begin:
%e A276664 n, prime(n), a(n)\  solutions (x, y)
%e A276664 1,   2,       2:   (0, 0), (1, 0)
%e A276664 2,   3,       1:   (1, 0)
%e A276664 3,   5,       6:   (0, 1), (0, 4), (1, 0),
%e A276664                    (3, 1), (3, 4), (4, 0)
%e A276664 4,   7,       9:   (1, 0), (2, 1), (2, 6),
%e A276664                    (4, 2), (4, 5), (5, 2),
%e A276664                    (5, 5), (6, 2), (6, 5)
%o A276664 (Ruby)
%o A276664 require 'prime'
%o A276664 def A276664(n)
%o A276664   ary = []
%o A276664   Prime.take(n).each{|p|
%o A276664     a = Array.new(p, 0)
%o A276664     (0..p - 1).each{|i| a[(i * i) % p] += 1}
%o A276664     ary << (0..p - 1).inject(0){|s, i| s + a[(i * i * i - i * i + 4 * i - 4) % p]}
%o A276664   }
%o A276664   ary
%o A276664 end
%K A276664 nonn
%O A276664 1,1
%A A276664 _Seiichi Manyama_, Sep 12 2016