Let $k$ be a field with $\operatorname{char}(k) \ne 2,3$, and let $a,b \in k$. Suppose that the projective plane cubic $E \subset \mathbb{P}^2_k$ defined by
is nonsingular. Then the point $O = [0:1:0] \in E(k)$ is a flex point of $E$; equivalently, the tangent line to $E$ at $O$ intersects $E$ at $O$ with intersection multiplicity $3$.