On elliptic curves whose $3$-torsion subgroup splits as $\mu_3 \oplus \mathbb Z/3\mathbb Z$

Masaya Yasuda

doi:10.4134/CKMS.2012.27.3.497

Communications of the
Korean Mathematical Society CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

QR
Code

Article

HOME

ALL ARTICLES

View

Commun. Korean Math. Soc. 2012; 27(3): 497-503

Printed September 1, 2012

https://doi.org/10.4134/CKMS.2012.27.3.497

On elliptic curves whose $3$-torsion subgroup splits as $\mu_3 \oplus \mathbb Z/3\mathbb Z$

Masaya Yasuda

4-1-1, Kamikodanaka, Nakahara-ku

Abstract

Go to

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