A basic question on the algebraic dimension of an algebraic variety

Question

Let $W,V\subset \mathbb{C}^{2n}$ be irreducible varieties both of dimension $n$ defined as the $0$-sets of some prime ideals $I$, $J$ on $\mathbb{Q}[x_1,\dots,x_{2n}]$, ideals which remains prime in $\mathbb{C}[x_1,\dots,x_{2n}]$. Assume $p=(\lambda_1,\dots,\lambda_{2n})$ is an isolated point of $V\cap W$. Is the transcendence degree of the numbers $\lambda_1,\dots,\lambda_{2n}$ over $\mathbb{Q}$ equal to $n$? i.e. equal to the geometric dimension of $V$? Or in general this is not the case?