Let $C$ be projective algebraic curve. A point $P\in C$ is called a flex point if the tangent line $l_P$ of $C$ in $P$ intersect $C$ with multiplicity almost 3.
Are flex points invariant under isomorphism of algebraic curves?
This doubt come from the fact that, as I understand, every elliptic curve $(E,O)$ is isomorphic to an elliptic curve $(E',O_\infty)$, where $E':y^2=4x^3-ax-b$ and $O_\infty=[0:1:0]$ (this is the Weierstrass equation of the elliptic curve). Now, the problem is that an isomorphisms in the category of elliptic curves must preserve the specified point, but $O$ is an arbitrary point of $E$ and $O_\infty$ is a flex point in $E'$.
Hope someone can clarify this to me. Thanks in advance.