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:

 $\mathbb{Z}/N\mathbb{Z}\quad\text{ with }1
 $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2N\mathbb{Z}\text{ with }0

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

CURVE TORSION SUBGROUP
$y^{2}=x^{3}-2$ trivial $\mathcal{O}$
$y^{2}=x^{3}+8$ $\mathbb{Z}/2\mathbb{Z}$ $[[-2,0]]$
$y^{2}=x^{3}+4$ $\mathbb{Z}/3\mathbb{Z}$ $[[0,2]]$
$y^{2}=x^{3}+4x$ $\mathbb{Z}/4\mathbb{Z}$ $[[2,4]]$
$y^{2}-y=x^{3}-x^{2}$ $\mathbb{Z}/5\mathbb{Z}$ $[[0,1]]$
$y^{2}=x^{3}+1$ $\mathbb{Z}/6\mathbb{Z}$ $[[2,3]]$
$y^{2}=x^{3}-43x+166$ $\mathbb{Z}/7\mathbb{Z}$ $[[3,8]]$
$y^{2}+7xy=x^{3}+16x$ $\mathbb{Z}/8\mathbb{Z}$ $[[-2,10]]$
$y^{2}+xy+y=x^{3}-x^{2}-14x+29$ $\mathbb{Z}/9\mathbb{Z}$ $[[3,1]]$
$y^{2}+xy=x^{3}-45x+81$ $\mathbb{Z}/10\mathbb{Z}$ $[[0,9]]$
$y^{2}+43xy-210y=x^{3}-210x^{2}$ $\mathbb{Z}/12\mathbb{Z}$ $[[0,210]]$
$y^{2}=x^{3}-4x$ $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$ $[[2,0],[0,0]]$
$y^{2}=x^{3}+2x^{2}-3x$ $\mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$ $[[3,6],[0,0]]$
$y^{2}+5xy-6y=x^{3}-3x^{2}$ $\mathbb{Z}/6\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$ $[[-3,18],[2,-2]]$
$y^{2}+17xy-120y=x^{3}-60x^{2}$ $\mathbb{Z}/8\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$ $[[30,-90],[-40,400]]$
Title examples of torsion subgroups of elliptic curves ExamplesOfTorsionSubgroupsOfEllipticCurves 2013-03-22 14:22:44 2013-03-22 14:22:44 alozano (2414) alozano (2414) 5 alozano (2414) Example msc 14H52 ArithmeticOfEllipticCurves