Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 125 / 3 / b / Solution

The Hessian determinant of $u^3+v^3+w^3$ is a nonzero scalar multiple of $uvw$. Hence the inflection points areThere are nine of them. Since $O_C=(1:-1:0)$ is an inflection point, these are exactly $C[3]$. They are all defined overso $C(K)[3]\cong (\mathbb{Z}/3\mathbb{Z}{)}^2$.

Solved by gpt-5.6-sol high.