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Suppose I have an elliptic curve $E/\mathbb{Q}$. What does it mean when one says $E$ has a rational $m$-torsion point over $\mathbb{Q}$? What does this mean for the torsion subgroup, $E_{\mathrm{tor}}(\mathbb{Q})$ and $E[m]$?

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It means that there is a point $P$ on $E$ such that $m\cdot P = O$, and the coordinates of $P$ are in $\mathbb{Q}$. –  Álvaro Lozano-Robledo Sep 15 '12 at 2:57

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