How to calculate the $p$-torsion points of an elliptic curve?
Consider the elliptic curve $E: \ y^2=x^3-5$ over $\mathbb{Q}$.
Then it is given that $E[2]=\{0,~(\sqrt[3]{5},0),~(\zeta_2 \sqrt[3]{5},0),~(\zeta_3^2 \sqrt[3]{5},0) \}$. see for instance Page $2$ here
Clearly these points satisfy the elliptic curve though they do not belong to $\mathbb{Q}$.
But I didn't see how these are $2$-torsion points.
Can you help me to explain?
If $P=(x,y)$ be a $2$-torsion point of $E: \ y^2=x^3-5$, then $2P=0$.
Also, What are the $3$-torsion points ?
Do Pari/gp calculate torsion points ?
Best Answer
As hinted by @Somos, but denied by @hunter, finding $3$-torsion points is easy. Consider the following sequence of logical equivalences, in which I use $T_EP$ to mean the tangent line to the elliptic curve at $P$, and I call the point at infinity $\Bbb O$: \begin{align} P\text{ is $3$-torsion}&\Leftrightarrow[3](P)=\Bbb O\\ &\Leftrightarrow[2](P)=-P\\ &\Leftrightarrow T_EP\text{ has its third intersection with $E$ at the point symmetric to $-P$}\\ &\Leftrightarrow T_EP\text{ has its third intersection with $E$ at $P$}\\ &\Leftrightarrow T_EP\text{ makes $3$-fold contact with $E$ at $P$}\\ &\Leftrightarrow\text{ $P$ is an inflection point of $E$ .} \end{align} Note that this accords with the well-known fact that the point at infinity is an inflection point of $E$ .