How to calculate the $p$-torsion points of an elliptic curve? 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 ?
 A: By definition of the group law, in Weierstrass form, a $2$-torsion point is a point where the graph has a vertical tangent.  This is equivalent to the $y$-coordinate being zero (by the symmetry about the $x$-axis).
A: 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 $E$ .
A: The LMFDB is very useful for these kind of questions. In particular, the
Elliptic curve 10800.be1
defined by $\,y^2 = x^3-5\,$ is the one you asked about. At the bottom of
the web page is a section "Growth of torsion in number fields" The first
entry is the curve over $\,K=\mathbb{Q}(\sqrt{-5})\,$ with the torsion
group listed as $\,\mathbb{Z}/3\mathbb{Z}\,$ which implies that there is
a $3$-torsion point. We guess that such a point is $\,P=(0,\sqrt{-5})\,$
and in fact $\,-P=P+P=(0,-\sqrt{-5}).\,$ You can do the simple
calculations with PARI/GP:
al = Mod(x, x^4+5); /* al^4 = -5 */
E = ellinit([0,0,0,0,al^2]); /* y^2 = x^3-5 */
P = [0,al]; /* (0,sqrt(-5)) */
ellisoncurve(E, P) /* ==1 True */
-P == ellmul(E, P, 2) /* ==1 True 2*P = (0,-sqrt(-5)) */

