Notation. Let $p$ be a prime number, $K$ a finite extension of $\mathbb{Q}_p$ and $E|K$ an elliptic curve which has good reduction. The discriminant $d_{E|K}$ of $E|K$ is an element of the multiplicative group $\mathfrak{o}^\times/\mathfrak{o}^{\times12}$, where $\mathfrak{o}$ is the ring of integers of $K$.
Question. Does the order of $d_{E|K}$ as an element of $\mathfrak{o}^\times/\mathfrak{o}^{\times12}$ show up somewhere ? Is it related to some other invariant of $E|K$ ?
Background. $E$ can be defined over $K$ by a minimal cubic
$f=y^2+a_1xy+a_3y-x^3-a_2x^2-a_4x-a_6=0$;f=y^2+a_1xy+a_3y-x^3-a_2x^2-a_4x-a_6=0,\ \ (a_i\in\mathfrak{o})$;
its discriminant $d_f$ is in $\mathfrak{o}^\times$ (because $E$ has good reduction). If we replace $f$ by another minimal cubic $g$ defining $E$, then $d_f$ gets replaced by $d_g=u^{12}d_f$ for some $u\in\mathfrak{o}^\times$. So the class $d_{E|K}$ of $d_f$ in $\mathfrak{o}^\times/\mathfrak{o}^{\times12}$ depends only on $E|K$, not on the choice of a minimal cubic defining $E$. It can be shown that every class in the finite group $\mathfrak{o}^\times/\mathfrak{o}^{\times12}$ is the discriminant of some good-reduction elliptic curve.
Addendum. As Qing Liu remarks, one may ask, given an elliptic curve $E$ over a finite extension $k|\mathbb{F}_p$, whether the order of the discriminant $d_{E|k}\in k^\times/k^{\times12}$ shows up somewhere. When $p\neq2,3$, the two questions are equivalent.
