For a short Weierstrass equation $E:y^2=x^3+ax+b$ in characteristic different from two, the <invariant differential on an elliptic curve>
$$
\omega=\frac{dx}{2y}
$$
is nonzero and regular, including at $O$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 125 / 3 / a / Solution

{wiki=Invariant_differential}

On $y^2=x^3+ax+b$ in characteristic other than two, the differential
$$
\omega=\frac{dx}{2y}
$$
extends regularly and without zeros across the point at infinity. Translation by every point preserves it.


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 Incoming links: Invariant differential on an elliptic curve