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A272202 Number of solutions of the congruence y^2 == x^3 - 1 (mod p) as p runs through the primes.

%I A272202 #20 Apr 07 2020 22:10:28
%S A272202 2,3,5,3,11,11,17,27,23,29,27,47,41,51,47,53,59,47,51,71,83,75,83,89,
%T A272202 83,101,123,107,107,113,147,131,137,123,149,147,143,171,167,173,179,
%U A272202 155,191,191,197,171,195,195,227,251,233,239,227,251,257,263,269,243,251,281
%N A272202 Number of solutions of the congruence y^2 == x^3 - 1 (mod p) as p runs through the primes.
%C A272202 In the Martin and Ono reference, in Theorem 2, this elliptic curve appears in the last column, starting with Conductor 144, as a strong Weil curve for the weight 2 newform  eta^{12}(12*z) / (eta^4(6*z) * eta^4(24*z)), symbolically 12^{12} 6^{-4} 24^{-4}, with Im(z) > 0, and the Dedekind eta function. See A187076 which gives the q-expansion (q = exp(2*Pi*i*z)) of exp(-Pi*i*z/3)* eta(2*z)^{12} / (eta^4(z)*eta^4(4*z)). For the q-expansion of 12^{12} 6^{-4} 24^{-4} one has a leading zero and 5 interspersed 0's: 0,1,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,-8,...
%C A272202 The discriminant of this elliptic curve is -3^3 = -27.
%C A272202 For the elliptic curve y^2 == x^3 + 1 (mod prime(n)) see A000727, A272197, A272198, A272200 and A272201.
%H A272202 Seiichi Manyama, <a href="/A272202/b272202.txt">Table of n, a(n) for n = 1..10000</a>
%H A272202 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 A272202 a(n) gives the number of solutions of the congruence y^2 == x^3 - 1 (mod prime(n)), n >= 1.
%e A272202 The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used. The solutions (x, y) of y^2 == x^3 - 1 (mod prime(n)) begin:
%e A272202 n, prime(n), a(n)\ solutions (x, y)
%e A272202 1,    2,      2:   (0, 1), (1, 0)
%e A272202 2,    3,      3:   (1, 0), (2, 1), (2, 2)
%e A272202 3,    5,      5:   (0, 2), (0, 3), (1, 0),
%e A272202                    (3, 1), (3, 4)
%e A272202 4,    7,      3:   (1, 0), (2, 0), (4, 0)
%e A272202 5,    11,    11:   (1, 0), (3, 2), (3, 9),
%e A272202                    (5, 5), (5, 6), (7, 1),
%e A272202                    (7, 10), (8, 4), (8, 7),
%e A272202                    (10, 3), (10, 8)
%e A272202 ...
%Y A272202 Cf. A000040, A187076, A272203.
%K A272202 nonn,easy
%O A272202 1,1
%A A272202 _Wolfdieter Lang_, May 05 2016