A127309 a(n) = |E(GF(p))| - (p+1) where E(GF(p)) is the group of rational points on the elliptic curve E: y^2 + y = x^3 - x^2 mod p and the prime p is p(n) or p(n+1) according as n < 5 or n >= 5.

2, 1, -1, 2, -4, 2, 0, 1, 0, -7, -3, 8, 6, -8, 6, -5, -12, 7, 3, -4, 10, 6, -15, 7, -2, 16, -18, -10, -9, -8, 18, 7, -10, 10, -2, 7, -4, 12, 6, 15, -7, -17, -4, 2, 0, -12, -19, -18, -15, -24, 30, 8, 23, 2, -14, -10, 28, 2, 18, -4, -24, -8, -12, 1, -13, -7, 22, -28

Offset

1,1

Comments

E is singular over GF(p(5)) = GF(11) so we take p != 11.

Hasse proved that |a(n)| <= 2*sqrt(p) where p is p(n) or p(n+1) according as n < 5 or n >= 5.

Elkies proved that a(n) = 0 for infinitely many n.

References

N. Koblitz, Introduction to Elliptic Curves and Modular Forms. New York: Springer-Verlag, 1993.

J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Math., vol. 106, Springer-Verlag, Berlin and New York, 1986.

Links

Robin Visser, Table of n, a(n) for n = 1..10000

S. Fermigier, Collection of Links on Research Articles on Elliptic Curves and Related Topics

B. Mazur, The Structure of Error Terms in Number Theory and an Introduction to the Sato-Tate Conjecture, Current Events Bulletin, Amer. Math. Soc., 2007.

Formula

a(n) = -b(p) where q * Product_{k>=1} ((1 - q^k)*(1 - q^(11*k)))^2 = Sum_{k>=1} b(k)*q^k is the g.f. of A006571 and p is p(n) or p(n+1) according as n < 5 or n >= 5.

Example

q*Product_{k>=1} ((1 - q^k)*(1 - q^11k))^2 = q - 2q^2 - ..., so a(1) = -b(p(1)) = -b(2) = -(-2) = 2.

Prog

(Sage)
def a(n):
    if n < 5: p = Primes()[n-1]
    else: p = Primes()[n]
    E = EllipticCurve(GF(p), [0, -1, 1, 0, 0])
    return -E.trace_of_frobenius()  # Robin Visser, Jul 01 2023

Crossrefs

|E(GF(p))| is A127310. Cf. A000594, A006571, A127311.

Sequence in context: A293176 A297170 A359627 * A097853 A160266 A322134

Adjacent sequences: A127306 A127307 A127308 * A127310 A127311 A127312

Keyword

sign

Author

Jonathan Sondow, Jan 12 2007

Extensions

More terms from Robin Visser, Jul 01 2023

Status

approved