primes that ramify in division fields for hyperelliptic jacobians

Question

Let $C$ be a hyperelliptic curve $y^2=f(x)$ of genus $g\geq 2$ and $\Delta$ the discriminant of $f(x)$. Let $\ell>2$ be a prime that divides $\Delta$ to the order $e:=\operatorname{ord}_\ell(\Delta)\geq 1$. Let $n\neq 2, \ell$ be a prime number and $\mathbb{Q}(J[n])$ be the field fixed by the representation on the $n$-torsion of the Jacobian $J$ of $C$: $$ \bar{\rho}'_{J, n}: \operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) \rightarrow \operatorname{GSp}_{2g}(\mathbb{Z}/n\mathbb{Z})/\langle -\operatorname{Id}\rangle$$. Then are there some conditions on $e$, $n$ and $\ell$ that will guarantee that $\ell$ ramifies in $\mathbb{Q}(J[n])$. For instance, can one show that if $e$ is bounded above by a certain quantity depending on $\ell$ and $n$, then it is indeed ramified? What about when $e$ is small, like $e=1$?