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Let $E$ be the elliptic curve $$y^2=x(x-2)(x-10)$$ Silverman claims (Example. X. 1.5 p.315 Arithmetic of Elliptic Curves) that $E(\mathbb{Q})_{tors}$ injects into the reduction $\widetilde{E}(\mathbb{F}_3)$. I understand by VII.3.1 earlier in his book that the $m$-torsion for all $m$ prime to $3$ injects into $\widetilde{E}(\mathbb{F}_3)$ . So my question is about the 3-torsion. Why does $E$ have no $\mathbb{Q}$ points that are 3-torsion (i.e. $[3]P=0$)?

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In the geometric picture the $3$-torsion points are precisely the inflection points of the curve. That is to say, writing an equation for the elliptic curve as $y^2=f(x)$, the $3$-torsion points are the points on the curve with with $x$-coordinate satisfying $f'(x)=0$ and $f''(x)=0$.

In this case there is no inflection point because $f'(x)=3x^2-24x+20$ and $f''(x)=6x-24$ have no common zero, so there is no $3$-torsion.

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