Let $\tau : \mathbb{C} \to \mathbb{C}^2$ be the map $\tau(t) := (t^2, t^3)$. Show that $\tau$ defines an embedding map from $\mathbb{C}^*$ to $\mathbb{C}^2 \setminus{0}$. Is $\tau(\mathbb{C})$ a submanifold of $\mathbb{C}$?
Note that the definition of an embedding map is a map that is holomorphic (i.e. termwise holomorphic), injective and proper. Furthermore, a submanifold of $\mathbb{C}^2$ is defined to be the embedding map of some manifold into $\mathbb{C}^2$ such that each point has Jacobian of maximal rank.
It's clear that $\tau$ is an embedding map. I want to show that it is not a submanifold. Intuitively, by drawing out $\tau(\mathbb{C})$, one can see a "sharp point" at $(0,0)$, which should admit a singularity. Indeed, one can easily check that the Jacobian at $(0,0)$ is the zero matrix. However, this does not count as a proof, as we need to show that no such embedding map exists, not just $\tau$. I'm not sure how to proceed.
Any help is appreciated.
HINT: if it were a submanifold then around $(0,0)$ it would be the graph of a function of $x$ or a function of $y$. Check neither is true.