On elliptic curves whose $3$-torsion subgroup splits as $\mu_3 \oplus \mathbb Z/3\mathbb Z$
Commun. Korean Math. Soc. 2012 Vol. 27, No. 3, 497-503
Printed September 1, 2012
Masaya Yasuda
4-1-1, Kamikodanaka, Nakahara-ku
Abstract : In this paper, we study elliptic curves $E$ over $\mathbb Q$ such that the $3$-torsion subgroup $E[3]$ is split as $\mu_3 \oplus \mathbb Z/3\mathbb Z$. For a non-zero integer $m$, let $C_m$ denote the curve $x^3 + y^3 = m$. We consider the relation between the set of integral points of $C_m$ and the elliptic curves $E$ with $E[3] \simeq \mu_3 \oplus \mathbb Z/3\mathbb Z$.
Keywords : elliptic curves, torsion points, V\'elu's formula
MSC numbers : Primary 14H52; Secondary 11G05
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