This is *not* an answer to my question but a long comment, mostly directed at Professor Silverman's answer.

In fact it is thinking about Section VIII.8 of your book and especially your 1984 Mathematika [paper][1] 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 $\mathbb{Q}_p$ 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][2], 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 $[L:K]$, 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 curve $E$ ? 

For example, for which curves $E$ is $\delta_{E|K}$ trivial in $\mathfrak{o}^\times/\mathfrak{o}^{\times 12}$ ?


  [1]: http://ams.org/mathscinet/search/publdoc.html?extend=1&extend=1&pg1=IID&s1=162205&vfpref=html&r=114&mx-pid=804199
  [2]: http://ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=frohlich&s5=algebraic&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=8&mx-pid=113876