Let $K$ be a local field, complete with respect to a discrete valuation $\nu$ (for example, $K$ could be $\Rats_p$ , the field of $p$ -adic numbers, which is complete with respect to the $p$ -adic valuation).

Let $E/K$ be an elliptic curve defined over $K$ given by a Weierstrass equation $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ where $a_1, a_2, a_3,a_4,a_6$ are constants in $K$ . By a suitable change of variables, we may assume that $\nu(a_i)\geq 0$ . As it is pointed out in this entry, any other Weierstrass equation for $E$ is obtained by a change of variables of the form $$x=u^2x'+r,\quad y=u^3y'+su^2x'+t$$ with $u,r,s,t\in K$ and $u\neq 0$ . Moreover, by Proposition 2 in the same entry, the discriminants of both equations satisfy $\Delta=u^{12}\Delta'$ , so they only differ by a $12$ th power of a non-zero number in $K$ . Let us define a set: $$S=\{ \nu(\Delta) : \Delta \text{ is the discriminant of a Weierstrass eq. for $E$ and } \nu(\Delta)\geq 0\}$$ Since $\nu$ is a discrete valuation, the set $S$ is a set of non-negative integers, therefore it has a minimum value $m\in S$ . Moreover, by the remark above, $m$ satisfies $0\leq m <12$ and $m$ is the unique number $t\in S$ with $0\leq t < 12$ .

It follows from the discussion above that every elliptic curve over a local field $K$ has a minimal model over $K$ .

It can be shown that all elliptic curves over $\Rats$ have a global minimal model. However, this is not true over general number fields. There exist elliptic curves over a number field $F$ which do not have a global minimal model (i.e. any given model is not minimal at $\nu$ for every $\nu$ ).