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

Title & Authors
ON ELLIPTIC CURVES WHOSE 3-TORSION SUBGROUP SPLITS AS μ3 ⊕ℤ/3ℤ
Yasuda, Masaya;

Abstract
In this paper, we study elliptic curves E over $\small{\mathbb{Q}}$ such that the 3-torsion subgroup E[3] is split as $\small{{\mu}_3{\oplus}\mathbb{Z}/3{\mathbb{Z}}}$. For a non-zero intege $\small{m}$, let $\small{C_m}$ denote the curve $x^3+y^3 Keywords elliptic curves;torsion points;V$\small{\acute{e}}$lu`s formula; Language English Cited by References 1. G. Frey, Some remarks concerning points of finite order on elliptic curves over global fields, Ark. Mat. 15 (1977), no. 1, 1-19. 2. T. Hadano, Elliptic curves with a torsion point, Nagoya Math. J. 66 (1977), 99-108. 3. I. Miyawaki, Elliptic curves of prime power conductor with${\mathbb{Q}}\$-rational points of finite order, Osaka J. Math. 10 (1973), 309-323.

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J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Math. 106, Springer-Verlag, Berlin-Heidelberg New York, 1994.

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J. Velu, Isogenis entre courbs elliptiques, C. R. Acad. Sci. Paris Ser. A-B (1971), 238-241.