Take the tour ×  Let $E/K$ be an elliptic curve, where $K$ is a
complete local field with residue field $k$ and char$(k) = p$.
I'm trying to make sense of Kodaira symbols and Tate's
algorithm.  My current understanding is:  I$_0$ = good reduction
I$_n$ = multiplicative reduction with $\nu(j) = -n$.  I$_n^\ast$
= potential multiplicative reduction, eventually becoming I$_n$
in a field extension (so $\nu(j) = -n$).  I$_0^\ast$ = "non-
exceptional" potential good reduction.  II, II$^\ast$, III,
III$^\ast$, IV, IV$^\ast$ = "exceptional" potential good
reduction. These can only happen when the $j$-invariant is
equivalent to 0 or 1728 modulo $p$, or when $p = 2, 3$ (where
everything is more complicated...)  Is this correct? Silverman's
Advanced Topics in the Arithmetic of Elliptic Curves has a good
table of reduction types when $k$ is algebraically closed, but I
haven't been able to find something analogous for more general
fields which gives me an overview of the possibilities. Also,
why are Kodaira symbols named