A272196 - OEIS

A272196

Number of solutions of the congruence y^2 == x^3 - 4*x^2 + 16 (mod p) as p runs through the primes.

2, 4, 4, 9, 10, 9, 19, 19, 24, 29, 24, 34, 49, 49, 39, 59, 54, 49, 74, 74, 69, 89, 89, 74, 104, 99, 119, 89, 99, 104, 119, 149, 144, 129, 159, 149, 164, 159, 179, 179, 194, 174, 174, 189, 199, 199, 199, 204, 209, 214, 209, 269, 249, 274, 259, 249, 259, 299, 279, 299

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OFFSET

1,1

COMMENTS

This elliptic curve is discussed in the second Silverman reference. The present sequence is given in the rows named N_p in Table 45.6, p. 403. In the rows named a_p the p-defects prime(n) - a(n) are shown.

This sequence also gives the number of solutions of the congruences y^2 + y == x^3 - x^2 - 10*x - 20 (mod prime(n)) as well as y^2 + y == x^3 - x^2 (mod prime(n)) for n > 1 (cf. A376073). The first one is given in the Martin and Ono reference in Theorem 2, first row of the table, and the second one is given in the Frenkel reference, p. 84. (Of course, one could change the sign of y in both congruences.)

The modularity pattern for the elliptic curve y^2 = x^3 - 4*x^2 + 16 (and the ones mentioned in the previous comment and a comment below) is exhibited by the modular cusp form of weight 2 and level 11 (eta(z)*eta(11*z))^2, where eta is the Dedekind function, which in the q = exp(2*Pi*i*z), (Im(z) > 0) expansion has coefficients given in A006571 (with A006571(0) = 0). For all odd primes (2 is a bad prime), A002070(n) = A006571(prime(n)) = prime(n) - a(n), n >= 2, the p-defect. A006571(2) = -2, not 2-2 = 0. Note that the discriminant of this elliptic curve is -2^8*11 (sometimes -2^12*11 is used). Prime 11 is also bad for this curve, but A006571(11) = 1 = 11 - a(5) = 11 - 10. The curve y^2 + y = x^3 - x^2 - 10*x - 20 has discriminant -11^5 (see the first Silverman reference, pp. 46-48).

From Wolfdieter Lang, Jan 02 2017: (Start)

The congruence y^2 + y == x^3 - x^2 - 7820*x - 263580 (mod p) as p runs through the odd primes has the same number of solutions. See the Cremona link, N=11.

If b_n(Q) is the number of solutions of the Diophantine equation Q(x1,x2,x3,x4) = n with the quadratic form Q(x1,x2,x3,x4) = x1^2 + 4*(x2^2+x3^2+x4^2) + x1*x3 + 4*x2*x3 + 3*x2*x4 + 7*x3*x4 then the theta series delta(q;Q) = 1 + Sum_{n>=1} b_n(Q)*q^n equals (1/5)*E(q) + (18/5)*f(q) with the expansion coefficients of E(q) given by A185699 and those of f(q) = (eta(z)*eta(11*z))^2 with q = exp(2*Pi*i*z), (Im(z) > 0) given by A006571. See the Moreno-Wagstaff reference, pp. 245-246. b_n(Q), E(q) and f(q) are there denoted by a_n(Q), 12*E_{Chi0}(z) and f(z), respectively, and a missing n in the numerator of E_{Chi0}(z) has to be added (see A185699). (End)

REFERENCES

Edward Frenkel, Liebe und Mathematik, Springer, Spektrum, 2014, p. 84.

Carlos J. Moreno and Samuel S. Wagstaff, Jr., Sums of Squares of Integers, Chapman & Hall/CRC, Boca Raton, London, New York, pp. 246-247 (corrected).

J. H. Silverman, The Arithmetic of Elliptic Curves, Springer, 1986, pp. 46-48.

J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, Table 45.6, p. 403, Theorem 47.2, p. 413 (4th ed., Pearson 2014, Table 6, p. 369, Theorem 2, p. 383)

LINKS

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

J. E. Cremona, Algorithms for Modular Elliptic Curves.

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

Carlos J. Moreno and Samuel S. Wagstaff, Jr., Sums of Squares of Integers, Chapman & Hall/CRC, Boca Raton, London, New York, pp. 246-247.

J. H. Silverman, The Arithmetic of Elliptic Curves, Springer, 1986, pp. 46-48.

FORMULA

a(n) gives the number of solutions of the congruence y^2 == x^3 - 4*x^2 + 16 (mod prime(n)), n >= 1.

EXAMPLE

The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used. The solutions (x, y) of y^2 == x^3 - 4*x^2 + 16 (mod prime(n)) begin:

n, prime(n), a(n)\ solutions (x, y)

1, 2, 2: (0, 0), (1, 1)