I am wondering if the following holds or not.

Let A be an abelian variety of dimension $d\geq 1$ over $\mathbb{Q}$. Then there is a positive number c depending on d and A such that $[\mathbb{Q}(A[n]):\mathbb{Q}]\geq c^{w(n)} n^2$ where $w(n)$ is the number of distinct prime factors of n.

I know that it must hold for d=1(elliptic curve case). I guess that we can even have stronger inequality like $[\mathbb{Q}(A[n]):\mathbb{Q}]\geq c^{w(n)} |GSp_{2d}(Z/nZ)|$ ,but don't know how to prove it, and couldn't find any reference other than d=1 case.

Thanks, Sungjin Kim.