ON ELLIPTIC CURVES WHOSE 3-TORSION SUBGROUP SPLITS AS μ3⊕ℤ/3ℤ

Abstract

In this paper, we study elliptic curves E over <TEX>$\mathbb{Q}$</TEX> such that the 3-torsion subgroup E[3] is split as <TEX>${\mu}_3{\oplus}\mathbb{Z}/3{\mathbb{Z}}$</TEX>. For a non-zero intege <TEX>$m$</TEX>, let <TEX>$C_m$</TEX> denote the curve <TEX>$x^3+y^3=m$</TEX>. We consider the relation between the set of integral points of <TEX>$C_m$</TEX> and the elliptic curves E with <TEX>$E[3]{\simeq}{\mu}_3{\oplus}\mathbb{Z}/3{\mathbb{Z}}$</TEX>.