  Let $R$ be a complete discrete valuation ring with fraction field $K$ and
perfect residue field $k$ of characteristic $p>0$. Let $E/K$ be an elliptic
curve with a $K$-rational isogeny of prime degree $\ell$. In this article, we
study the possible Kodaira types of reduction that $E/K$ can have. We also
prove some related results for elliptic curves over $\mathbb{Q}$.
