 Divisibility proof in the Nagell–Lutz theorem

For an integral torsion point $(x,y)$ with $y\ne 0$, 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 inshows that $y^2\mid 4a^3+27b^2$.