Rational Points of Elliptic Curve y2=x3+k3

Abstract

Let E be an elliptic curve defined over the field of rational numbers ℚ. Let d be a square-free integer and let Ed be the quadratic twist of E determined by d. Mai, Murty and Ono have proved that there are infinitely many square-free integers d such that the rank of Ed(ℚ) is zero. Let E(k) denote the elliptic curve y2 = x3 + k. Then the quadratic twist E(1)d of E(1) by d is the elliptic curve [Formula: see text]. Let r = 1, 2, 5, 10, 13, 14, 17, 22. Ono proved that there are infinitely many square-free integers d ≡ r (mod 24) such that rank [Formula: see text], using the theory of modular forms. In this paper, we use the class number of quadratic field and Pell equation to describe these square-free integers k such that E(k3)(ℚ) has rank zero.