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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.
3
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.
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
KEYWORD
sign
AUTHOR
Jonathan Sondow, Jan 12 2007
EXTENSIONS
More terms from Robin Visser, Jul 01 2023
STATUS
approved