

A271230


Pdefects 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.


5



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
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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} (1q^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 pdefect.
The identical pdefects 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 etaquotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 48254856, see page 4852 Table I.
Yves Martin and Ken Ono, EtaQuotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 31693176.
Michael Somos, Index to Yves Martin's list of 74 multiplicative etaquotients and their Anumbers
Haode Yan, Yongbo Xia, Chunlei Li, Tor Helleseth, Maosheng Xiong and Jinquan Luo, The Differential Spectrum of the Power Mapping x^p^(n3)), 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



