This is not an answer but a clarification of my question.
In fact it is thinking about Section VIII.8 of your book and especially your 1984 Mathematika paper which led to the question.
As in your paper, there is an analogy with discriminants of number fields. Keeping to the purely local situation, let $K$ be a finite extension of the $p$-adics, with ring of integers $\mathfrak{o}$, and let $L$ be a finite extension of $K$. Then the discriminant $\delta_{L|K}$ of $L|K$ can be thought of, following Fröhlich, as an element of the group $K^\times/\mathfrak{o}^{\times 2}$.
When $L|K$ is unramified, $\delta_{L|K}$ is an element of $\mathfrak{o}^\times/\mathfrak{o}^{\times 2}$, and its order as an element of this group --- the only possibilities are $1$ and $2$ --- gives us the parity of $[L:K]$. More precisely, $\delta_{L|K}$ has order $1$ if $[L:K]$ is odd, order $2$ if $[L:K]$ is even.
Let's now return to our good-reduction elliptic curve $E$ over $K$, whose discriminant $\delta_{E|K}$ is an element of $\mathfrak{o}^\times/\mathfrak{o}^{\times 12}$. The question is, what does the order of the element $\delta_{E|K}$ in the said group --- the possibilities for the order being $1,2,3,4,6,12$ --- tell us about the curve $E$ ?
For example, for which curves $E$ is $\delta_{E|K}$ trivial in $\mathfrak{o}^\times/\mathfrak{o}^{\times 12}$ ?