In [this paper](https://arxiv.org/pdf/math/0007208.pdf) of Maja Volkov, the authur metions a number called "défaut de semi-stabilité" on page 9. It is defined as $\text{dst}(E)=\frac{12}{\text{pgcd} (12,v\_p(\Delta\_E))}$ where $E$ is an elliptic curve over $\mathbb{Q}\_p$ and "pgcd" is the greatest common divisor.


Then the authur says on the next paragraph that if $p>5$ and $E$ has potential good reduction over $\mathbb{Q}\_p$, then $e=\text{dst}(E)$ is prime to $p$ and $E$ has good reduction on an totally ramified extension of $\mathbb{Q}\_p$ of degree $e$.


So the question is how to prove the two propositions and I want to know if there are some reference about the relations between "défaut de semi-stabilité" and the reduction types of elliptic curves.


Thanks!



Explicit semi-stable theorem for elliptic curves over $p$-adic fields
More generally, let $K/\mathbb{Q}\_p$ be a finite extension with $p\ge5$, and let $E/K$ have potential good reduction. Then you can read off the Kodaira-Neron reduction type from the valuation of the minimal discriminant. This is in Tate's table in Antwerp IV, reproduced in my *Arithmetic of Elliptic Curves* book (Table 15.1) and many other places. Let $\pi$ be a uniformizer and $v$ the normalized valuation on $K$. Take a minimal Weierstrass equation for $E$, 
$$ E: Y^2 = X^3 + AX + B . $$
To get good reduction, we change variables by
$$ E: Y^2 = X^3 + \pi^{-4r}AX + \pi^{-6r}B \quad\text{with}\quad r = \min\left\{\frac14v(A),\frac16v(B)\right\}. $$
So there is a field of good reduction $K\_E^{\text{good}}$ given
by one of $K(A^{1/4})$ or $K(B^{1/6})$, from which one can read off the ramification index $e(K\_E^{\text{good}}/K)$. A case-by-case analysis yields complete information, as follows:
$$
\begin{array}{|c|c|c|} \hline
\text{Type} & II & III & IV & I\_0^\* & IV^\* &III^\* & II^\* \\ \hline
v(A) & \ge1 & 1 & \ge1 & =2~\text{or}~\ge2 & \ge3 & 3 & \ge4 \\ \hline
v(B) & 1 & \ge2 & 2 & \ge3~\text{or}~=3 & 4 & \ge5 & 5 \\ \hline
v(\Delta\_E) & 2 & 3 & 4 & 6 & 8 & 9 & 10 \\ \hline
K\_E^{\text{good}} & K(\pi^{1/6}) & K(\pi^{1/4}) & K(\pi^{2/6}) & K(\pi^{2/4})=K(\pi^{3/6}) & K(\pi^{4/6}) & K(\pi^{3/4}) & K(\pi^{5/6}) \\ \hline
e(K\_E^{\text{good}}/K) & 6 & 4 & 3 & 2 & 3 & 4 & 6 \\ \hline
12/\gcd(12,v(\Delta)) & 6 & 4 & 3 & 2 & 3 & 4 & 6 \\ \hline
\end{array}
$$


