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 normal model for $E/K$, which is a Weierstrass model satisfying
certain conditions that lead one to easily infer the local data of $E/K$. Our
main results provide necessary and sufficient conditions on the Weierstrass
coefficients of a normal 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 normal model for $E$.