A271230 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.

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, -16, 12, -2, 18, 8, 4, -6, 12, 14, 16, -2, -12, -24, 6, -12, 6, 0, 2, -18, -16, 20, 8, -12, 22, 10, 16, 18, -20, 2, 8, -10, -8, -26, 26

Offset

1,3

Comments

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.

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

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Haode Yan, Yongbo Xia, Chunlei Li, Tor Helleseth, Maosheng Xiong and Jinquan Luo, The Differential Spectrum of the Power Mapping x^p^(n-3)), arXiv:2108.03088 [cs.IT], 2021. See Table II p. 7.

Formula

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)).

a(n) = A271231(prime(n)), n >=1.

Example

See the example section of A271229.
n = 3, prime(3) = 5, A271229(5) = 7, a(3) = 5 - 7 = -2.

Crossrefs

Cf. A159819, A271229, A271231.

Sequence in context: A037035 A159984 A240697 * A112824 A195133 A308022

Adjacent sequences: A271227 A271228 A271229 * A271231 A271232 A271233

Keyword

sign,easy

Author

Wolfdieter Lang, Apr 18 2016

Status

approved