Let $E/\mathbb{Q} = E_{a,b}$ $$\displaystyle y^2 = x^3 + ax + b, a,b \in \mathbb{Z}$$ be an elliptic curve defined over the field of rational numbers, and let $n \geq 2$ be an integer. Let $K_n$ be the $n$-torsion field of $E$, namely the field obtained by adjoining all of the coordinates of $n$-torsion points of $E$ to $\mathbb{Q}$. Let $p$ be a prime divisor of the discriminant $\Delta(E)$ of $E$. Does it follow that $K_n$ is ramified at $p$? If so, what can be said about the splitting behaviour of $p$ in $K_n$?

Let $E/\mathbb{Q} = E_{a,b}$ $$\displaystyle y^2 = x^3 + ax + b, a,b \in \mathbb{Z}$$ be an elliptic curve defined over the field of rational numbers, and let $n \geq 3$ be an integer. Let $K_n$ be the $n$-torsion field of $E$, namely the field obtained by adjoining all of the coordinates of $n$-torsion points of $E$ to $\mathbb{Q}$. Let $p$ be a prime divisor of the discriminant $\Delta(E)$ of $E$. Does it follow that $K_n$ is ramified at $p$? If so, what can be said about the splitting behaviour of $p$ in $K_n$?