examples of torsion subgroups of elliptic curvesExamplesOfTorsionSubgroupsOfEllipticCurves2004-05-25 15:22:482004-05-26 10:14:42ExampleMazur's theorem on torsion of elliptic curves% this is the default PlanetMath preamble. as your knowledge
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% Some sets
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\newcommand{\Rats}{\mathbb{Q}}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:
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
{\bf CURVE} & {\bf TORSION SUBGROUP} & {\bf GENERATORS} \\
\hline
$y^2=x^3-2$ & trivial & $\mathcal{O}$ \\
$y^2=x^3+8$ & $\Ints/2\Ints$ & $[[-2,0]]$ \\
$y^2=x^3+4$ & $\Ints/3\Ints$ & $[[0,2]]$ \\
$y^2=x^3+4x$ & $\Ints/4\Ints$ & $[[2,4]]$ \\
$y^2-y=x^3-x^2$ & $\Ints/5\Ints$ & $[[0,1]]$ \\
$y^2=x^3+1$ & $\Ints/6\Ints$ & $[[2,3]]$ \\
$y^2=x^3-43x+166$ & $\Ints/7\Ints$ & $[[3,8]]$ \\
$y^2+7xy=x^3+16x$ & $\Ints/8\Ints$ & $[[-2,10]]$ \\
$y^2+xy+y=x^3-x^2-14x+29$ & $\Ints/9\Ints$ & $[[3,1]]$ \\
$y^2+xy=x^3-45x+81$ & $\Ints/10\Ints$ & $[[0,9]]$ \\
$y^2+43xy-210y=x^3-210x^2$ & $\Ints/12\Ints$ & $[[0,210]]$ \\
$y^2=x^3-4x$ & $\Ints/2\Ints \oplus \Ints/2\Ints$ & $[[2, 0], [0, 0]]$ \\
$y^2=x^3+2x^2-3x$ & $\Ints/4\Ints \oplus \Ints/2\Ints$ & $[[3,6],[0,0]]$ \\
$y^2+5xy-6y=x^3-3x^2$ & $\Ints/6\Ints \oplus \Ints/2\Ints$ & $[[-3, 18], [2, -2]]$ \\
$y^2 +17xy -120y=x^3 -60x^2$ & $\Ints/8\Ints \oplus \Ints/2\Ints$ & $[[30, -90], [-40, 400]]$ \\
\hline
\end{tabular}
\end{center}