I have a simple cubic curve, say $y^2 = x^3 - x.$
Is there a simple way to find a small finite subgroup of points lying on this curve? (with respect to the elliptic curve group law.)
Otherwise, does anyone know of any collection of explicit examples of such small finite subgroups?
Mazur's theorem shows that given an elliptic curve defined over the rationals, the only possible torsion subgroups are the following:
$$\mathbb{Z}/N\mathbb{Z}\quad \text{ with } 1\leq N\leq 10\ \text{ or }\ N=12,\ \text{ or} $$
$$\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2N\mathbb{Z} \quad \text{ with } 1\leq N\leq 4.$$
In your example, the torsion subgroup of $y^2=x^3-x$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, generated by $[-1,0]$ and $[0,0]$. There are several ways to compute the torsion subgroup; the Nagell-Lutz theorem is one effective way. The theorem says the following. Let $E: y^2=f(x)=x^3+Ax+B$, with $A,B\in\mathbb{Z}$ and $D=4A^3+27B^2\neq 0$. Let $P=[x_0,y_0]\in E(\mathbb{Q})$ be a point of finite order. Then, $x_0,y_0$ are integers. Moreover:
Either $f(x_0)=0$, or
$y_0^2$ is a divisor of $D=4A^3+27B^2$.
In your case, $E: y^2=x^3-x$ has $A=-1$, $B=0$, and $D=-4$. If $f(x)=0$, then $x=0,1$ or $-1$, which correspond to the points $[0,0]$, $[1,0]$, and $[-1,0]$. If $y_0^2$ is a divisor of $D=-4$, then either $y_0=\pm 1$ or $y_0=\pm 2$. But $x^3-x-1=0$ and $x^3-x-4=0$ are irreducible over $\mathbb{Q}$, so there are no other rational torsion points. Hence the torsion subgroup of $E(\mathbb{Q})$ is formed by $[0,0]$, $[1,0]$, and $[-1,0]$, and the point at infinity $[0,1,0]$.
Here we show examples of curves with the torsion subgroups mentioned above:
\begin{array}{ccc} \hline \text{CURVE} & \text{TORSION SUBGROUP} & \text{GENERATORS} \\ \hline y^2=x^3-2 & trivial & \mathcal{O} \\ y^2=x^3+8 & \mathbb{Z}/2\mathbb{Z} & [[-2,0]] \\ y^2=x^3+4 & \mathbb{Z}/3\mathbb{Z} & [[0,2]] \\ y^2=x^3+4x & \mathbb{Z}/4\mathbb{Z} & [[2,4]] \\ y^2-y=x^3-x^2 & \mathbb{Z}/5\mathbb{Z} & [[0,1]] \\ y^2=x^3+1 & \mathbb{Z}/6\mathbb{Z} & [[2,3]] \\ y^2=x^3-43x+166 & \mathbb{Z}/7\mathbb{Z} & [[3,8]] \\ y^2+7xy=x^3+16x & \mathbb{Z}/8\mathbb{Z} & [[-2,10]] \\ y^2+xy+y=x^3-x^2-14x+29 & \mathbb{Z}/9\mathbb{Z} & [[3,1]] \\ y^2+xy=x^3-45x+81 & \mathbb{Z}/10\mathbb{Z} & [[0,9]] \\ y^2+43xy-210y=x^3-210x^2 & \mathbb{Z}/12\mathbb{Z} & [[0,210]] \\ y^2=x^3-4x & \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} & [[2, 0], [0, 0]] \\ y^2=x^3+2x^2-3x & \mathbb{Z}/4\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} & [[3,6],[0,0]] \\ y^2+5xy-6y=x^3-3x^2 & \mathbb{Z}/6\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} & [[-3, 18], [2, -2]] \\ y^2 +17xy -120y=x^3 -60x^2 & \mathbb{Z}/8\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} & [[30, -90], [-40, 400]] \\ \hline \end{array}