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Mazur's theorem shows that given an elliptic curve defined over the rationals, the only possible torsion subgroups are the following:
$$\Ints/N\Ints \quad \text{ with } 1<N<11 \text{ or } N=12$$
$$\Ints/2\Ints \oplus \Ints/2N\Ints \text{ with } 0<N<5$$
Here we show examples of curves with the torsion subgroups mentioned above:
| CURVE |
TORSION SUBGROUP |
GENERATORS |
| $y^2=x^3-2$ |
trivial |
$\mathcal{O}$ |
| $y^2=x^3+8$ |
$\Ints/2\Ints$ |
$[[-2,0]]$ |
| $y^2=x^3+4$ |
$\Ints/3\Ints$ |
$[[0,2]]$ |
| $y^2=x^3+4x$ |
$\Ints/4\Ints$ |
$[[2,4]]$ |
| $y^2-y=x^3-x^2$ |
$\Ints/5\Ints$ |
$[[0,1]]$ |
| $y^2=x^3+1$ |
$\Ints/6\Ints$ |
$[[2,3]]$ |
| $y^2=x^3-43x+166$ |
$\Ints/7\Ints$ |
$[[3,8]]$ |
| $y^2+7xy=x^3+16x$ |
$\Ints/8\Ints$ |
$[[-2,10]]$ |
| $y^2+xy+y=x^3-x^2-14x+29$ |
$\Ints/9\Ints$ |
$[[3,1]]$ |
| $y^2+xy=x^3-45x+81$ |
$\Ints/10\Ints$ |
$[[0,9]]$ |
| $y^2+43xy-210y=x^3-210x^2$ |
$\Ints/12\Ints$ |
$[[0,210]]$ |
| $y^2=x^3-4x$ |
$\Ints/2\Ints \oplus \Ints/2\Ints$ |
$[[2, 0], [0, 0]]$ |
| $y^2=x^3+2x^2-3x$ |
$\Ints/4\Ints \oplus \Ints/2\Ints$ |
$[[3,6],[0,0]]$ |
| $y^2+5xy-6y=x^3-3x^2$ |
$\Ints/6\Ints \oplus \Ints/2\Ints$ |
$[[-3, 18], [2, -2]]$ |
| $y^2 +17xy -120y=x^3 -60x^2$ |
$\Ints/8\Ints \oplus \Ints/2\Ints$ |
$[[30, -90], [-40, 400]]$ |
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