PlanetMath: examples of torsion subgroups of elliptic curves

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examples of torsion subgroups of elliptic curves

(Example)

Mazur's theorem shows that given an elliptic curve defined over the rationals, the only possible torsion subgroups are the following:

$\displaystyle \mathbb{Z}/N\mathbb{Z}$ with $\displaystyle 1<N<11$ or $\displaystyle N=12$

$\displaystyle \mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2N\mathbb{Z}$ with $\displaystyle 0<N<5$

Here we show examples of curves with the torsion subgroups mentioned above:

CURVE	TORSION SUBGROUP	GENERATORS
	trivial	$\mathcal{O}$
	$\mathbb{Z}/2\mathbb{Z}$
	$\mathbb{Z}/3\mathbb{Z}$
	$\mathbb{Z}/4\mathbb{Z}$
	$\mathbb{Z}/5\mathbb{Z}$
	$\mathbb{Z}/6\mathbb{Z}$
	$\mathbb{Z}/7\mathbb{Z}$
	$\mathbb{Z}/8\mathbb{Z}$
	$\mathbb{Z}/9\mathbb{Z}$
	$\mathbb{Z}/10\mathbb{Z}$
	$\mathbb{Z}/12\mathbb{Z}$
	$\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$
	$\mathbb{Z}/4\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$
	$\mathbb{Z}/6\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$
	$\mathbb{Z}/8\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$