examples of torsion subgroups of elliptic curves

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

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

CURVE	TORSION SUBGROUP	GENERATORS
$y^2=x^3-2$	trivial	$𝒪$
$y^2=x^3+8$	$\mathbb{Z}/2\mathbb{Z}$	$[[-2,0]]$
$y^2=x^3+4$	$\mathbb{Z}/3\mathbb{Z}$	$[[0,2]]$
$y^2=x^3+4x$	$\mathbb{Z}/4\mathbb{Z}$	$[[2,4]]$
$y^2-y=x^3-x^2$	$\mathbb{Z}/5\mathbb{Z}$	$[[0,1]]$
$y^2=x^3+1$	$\mathbb{Z}/6\mathbb{Z}$	$[[2,3]]$
$y^2=x^3-43x+166$	$\mathbb{Z}/7\mathbb{Z}$	$[[3,8]]$
$y^2+7xy=x^3+16x$	$\mathbb{Z}/8\mathbb{Z}$	$[[-2,10]]$
$y^2+xy+y=x^3-x^2-14x+29$	$\mathbb{Z}/9\mathbb{Z}$	$[[3,1]]$
$y^2+xy=x^3-45x+81$	$\mathbb{Z}/10\mathbb{Z}$	$[[0,9]]$
$y^2+43xy-210y=x^3-210x^2$	$\mathbb{Z}/12\mathbb{Z}$	$[[0,210]]$
$y^2=x^3-4x$	$\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$	$[[2,0],[0,0]]$
$y^2=x^3+2x^2-3x$	$\mathbb{Z}/4\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$	$[[3,6],[0,0]]$
$y^2+5xy-6y=x^3-3x^2$	$\mathbb{Z}/6\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$	$[[-3,18],[2,-2]]$
$y^2+17xy-120y=x^3-60x^2$	$\mathbb{Z}/8\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$	$[[30,-90],[-40,400]]$

Title	examples of torsion subgroups of elliptic curves
Canonical name	ExamplesOfTorsionSubgroupsOfEllipticCurves
Date of creation	2013-03-22 14:22:44
Last modified on	2013-03-22 14:22:44
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	5
Author	alozano (2414)
Entry type	Example
Classification	msc 14H52
Related topic	ArithmeticOfEllipticCurves