Let $N \ge 5$ be a prime number and $E/ \mathbb{Q}_N$ be an elliptic curve with additive reduction. Then it is easy to see that there exists a finite extension $K$ over which $E$ has stable reduction.

I want to show that we can choose $K$ so that the ramification index of $K/ \mathbb{Q}_N \le 6$.

The proof of VII 5.5 of Silverman’s AEC does not include that.