On the Kodaira types of elliptic curves with potentially good supersingular reduction

Let $O_K$ be a Henselian discrete valuation domain with field of fractions K. Assume that $O_K$ has algebraically closed residue field k. Let $E/K$ be an elliptic curve with additive reduction. The semi-stable reduction theorem asserts that there exists a minimal extension $L/K$ such that the base change $E_L/L$ has semi-stable reduction.

It is natural to wonder whether specific properties of the semi-stable reduction and of the extension $L/K$ impose restrictions on what types of Kodaira type the special fiber of $E/K$ may have. In this paper we study the restrictions imposed on the reduction type when the extension $L/K$ is wildly ramified of degree 2, and the curve $E/K$ has potentially good supersingular reduction. We also analyze the possible reduction types of two isogenous elliptic curves with these properties.