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Abstract

\def\Qset{{\sym Q}}\def\Knr#1{{#1_{nr}}}% Let $K$ be a finite extension of $\Qset_2$ complete with a discrete valuation~$v$, $\overline{K}$ an algebraic closure of~$K$, and $\Knr{K}$ its maximal unramified subextension. Let $E$ be an elliptic curve defined over $K$ with additive reduction over $K$ and having an integral modular invariant~$j$. There exists a~smallest extension $L$ of $\Knr{K}$ over which $E$ has good reduction. For some congruences modulo~$12$ of the valuation $v(j)$ of $j$, we give the degree of the extension $L/\Knr{K}$. When $K$ is a quadratic ramified extension of $\Qset_2$, we determine explicitly this degree in terms of the coefficients of a~Weierstrass equation of~$E$.