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%I #23 Aug 09 2021 12:24:59
%S 0,1,-2,0,-4,-2,2,4,8,6,-8,6,-6,-4,0,-2,-4,-2,4,-8,10,8,4,-6,2,-18,
%T -16,12,-2,18,8,4,-6,12,14,16,-2,-12,-24,6,-12,6,0,2,-18,-16,20,8,-12,
%U 22,10,16,18,-20,2,8,-10,-8,-26,26
%N P-defects p - N(p) of the congruence y^2 == x^3 + x^2 + x (mod p) for primes p, where N(p) is the number of solutions.
%C The modularity pattern series is the expansion of the (corrected) Nr. 54 modular cusp form of weight 2 and level N=48 given in the table 1 of the Martin reference, i.e., (eta(4*z) * eta(12*z)^4 / (eta(2*z) * eta(6*z) * eta(8*z) * eta(24*z)) in powers of q = exp(2*Pi*i*z), with Im(z) > 0, where i is the imaginary unit. Here eta(z) = q^{1/24}*Product_{n>=1} (1-q^n) is the Dedekind eta function. See A271231 for this expansion. Note that also for the possibly bad prime 2 and the bad prime 3 (the discriminant of this elliptic curve is -3) this expansion gives the correct p-defect.
%C The identical p-defects occur for the elliptic curve y^2 = x^3 + x^2 - 4*x - 4 taken modulo prime(n). See the Martin and Ono reference, p. 3173, row Conductor 48, and A271231 (checked up to prime(100) = 541). - _Wolfdieter Lang_, Apr 21 2016
%H Seiichi Manyama, <a href="/A271230/b271230.txt">Table of n, a(n) for n = 1..10000</a>
%H Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H Yves Martin and Ken Ono, <a href="http://www.ams.org/journals/proc/1997-125-11/S0002-9939-97-03928-2/">Eta-Quotients and Elliptic Curves</a>, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.
%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%H Haode Yan, Yongbo Xia, Chunlei Li, Tor Helleseth, Maosheng Xiong and Jinquan Luo, <a href="https://arxiv.org/abs/2108.03088">The Differential Spectrum of the Power Mapping x^p^(n-3))</a>, arXiv:2108.03088 [cs.IT], 2021. See Table II p. 7.
%F a(n) = prime(n) - A271229(n), n >= 1, where A271229(n) is the number of solutions of the congruence y^2 == x^3 + x^2 + x (mod prime(n)).
%F a(n) = A271231(prime(n)), n >=1.
%e See the example section of A271229.
%e n = 3, prime(3) = 5, A271229(5) = 7, a(3) = 5 - 7 = -2.
%Y Cf. A159819, A271229, A271231.
%K sign,easy
%O 1,3
%A _Wolfdieter Lang_, Apr 18 2016
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