The elliptic curves y2 = x(x - 1)(x - λ)

TEKCAN A.

Ars Combinatoria, vol.99, pp.519-529, 2011 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 99
  • Publication Date: 2011
  • Journal Name: Ars Combinatoria
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.519-529
  • Keywords: Elliptic curves over finite fields, Rank of elliptic curves, Rational points on elliptic curves
  • Bursa Uludag University Affiliated: Yes

Abstract

Let p be a prime number and let Fp be a finite field. In the first section, we give some preliminaries from elliptic curves over finite fields. In the second section we consider the rational points on the elliptic curves Ep,λ : y2 = x(x - 1)(x - λ) over Fp for primes p ≡ 3 (mod 4), where λ ≠ 0, 1. We proved that the order of Ep,λ over Fp is p + 1 if λ = 2, p+1/2 or p - 1. Later we generalize this result to F pn for any integer n ≥ 2. Also we obtain some results concerning the sum of x-and y-coordinates of all rational points (x, y) on Ep,λ over Fp. In the third section, we consider the rank of Eλ: y2 = x(x - 1)(x - λ) over Q.