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

Abstract

Abstract. In this paper, we study elliptic curves E over Q such that the 3-torsion subgroup E[3] is split as µ 3 ⊕ Z/3Z. For a non-zero integer m, let Cm denote the curve x 3 + y 3 = m. We consider the relation between the set of integral points of Cm and the elliptic curves E with E[3] ≃ µ 3 ⊕ Z/3Z.