Divisibility proof in the Nagell–Lutz theorem
= Divisibility proof in the Nagell–Lutz theorem
For an integral torsion point $(x,y)$ with $y\ne0$, the point $2P$ is integral. The tangent slope $(3x^2+a)/(2y)$ has integral square and is rational, hence is integral. Thus $y^2$ divides $(3x^2+a)^2$. Combining this with the curve equation in
$$
(3x^2+4a)(3x^2+a)^2-27(x^3+ax-b)(x^3+ax+b)=4a^3+27b^2
$$
shows that $y^2\mid4a^3+27b^2$.