Local data of elliptic curves under quadratic twist

Abstract

Let $K$ be the field of fractions of a complete discrete valuation ring with a perfect residue field. In this article, we investigate how the Tamagawa number of $E/K$ changes under quadratic twist. To accomplish this, we introduce the notion of a strongly-minimal model for an elliptic curve $E/K$, which is a minimal Weierstrass model satisfying certain conditions that lead one to easily infer the local data of $E/K$. Our main results provide explicit conditions on the Weierstrass coefficients of a strongly-minimal model of $E/K$ to determine the local data of a quadratic twist $E^{d}/K$. We note that when the residue field has characteristic $2$, we only consider the special case $K=\mathbb{Q}_{2}$. In this setting, we also determine the minimal discriminant valuation and conductor exponent of $E$ and $E^d$ from further conditions on the coefficients of a strongly-minimal model for $E$.