A way to read off $\deg \phi(P)$ directly from $P$ where $\phi:\mathbb{C}[x_1, \cdots, x_n]^{\mathfrak{S}_n}\to \mathbb{C}[e_1, \cdots, e_n]$?

Question

Let $P(x_1, \cdots, x_n)\in \mathbb{C}[x_1, \cdots, x_n]^{\mathfrak{S}_n}$ be a symmetric polynomial and $Q(e_1, \cdots, e_n)$ be its image under the isomorphism $\mathbb{C}[x_1, \cdots, x_n]^{\mathfrak{S}_n}\to \mathbb{C}[e_1, \cdots, e_n]$ where $e_i$ are the elementary symmetric polynomials. Suppose we know that $Q(e_1, \cdots, e_n)$ is of degree $d$. My question is

Is there a way one can read off $d$ directly from $P$?More directly I'm wondering if there is a differential operator $D$ in $x_1, \cdots, x_n$ such that $DP=dP$?