# Torsion points of an elliptic curve (example in Silverman)

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. $$[3]P=0$$)?

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.