Let $E/\mathbb{Q}_p$, $p\geq5$ be an elliptic curve with additive potentially good reduction. Then there is a unique, minimal, finite and totally ramified extension $K$ such that $E/K$ has good reduction.
Now, given that $E$ has potentially good reduction, it has good reduction if and only if the valuation of the discriminant is divisible by $12$. This means the inertia group has order dividing $12$. Since we have assumed that $p\geq5$, there is no wild inertia which means he inertia group is cyclic of order dividing $12$.
However, it seems that it cannot have order $12$ due to the characteristic polynomial of the Frobenius being quadratic and the minimal polynomial of a primitive $12th$ root of unity being quartic. Can anyone please elaborate on this and how that helps?