Rational points on the elliptic curve $y^2=x^3 -x$

Question

I've been trying to find rational points on the elliptic curve $y^2=x^3 -x$ but I can't find anything else apart from $(-1,0), (0,0), (1,0)$. I have like an 'edgy' proof that it may not be possible to find rational points on this curve(apart from the ones already mentioned), but I'm unsure about it. Can anyone help?

Answer 1

On the website SageMath Cell server. Switch the language to "GP". Enter the lines

ellfromeqn(-y^2 + x^3 - x);
print("torsion = ",elltors(E));
print("generators = ",ellgenerators(E));

Click on "Evaluate" and the output is

torsion = [4, [2, 2], [[-1, 0], [0, 0]]]
generators = []

This tells you that the order of the torsion group is $4$, which is the product of two cyclic groups, and two generators of the torsion group is $(-1,0)$ and $(0,0)$. The Mordell-Weil group of $E$ is trivial and thus has no generators.

You can also download PARI/GP and do the same thing on your computer. You could download a stand-alone Windows binary and execute it without any installation needed.