Primitive local Galois representations: An example (Q1174734)

Let \(K\) be a \(p\)-adic number field, \(G_ K\) its absolute Galois group and \(\rho\) a primitive, irreducible representation of \(G_ K\). The wild ramification subgroup \(\overline{G}_ 1\) of the projective representation corresponding to \(\rho\) is an abelian group. If \(\overline{G}_ 1\) has one jump at \(s=1\), then \textit{E.-W. Zink} [Lokale projektive Klassenkörpertheorie II, Math. Nachr. 114, 123-150 (1983; Zbl 0547.12012)] has computed the possible values of the Swan conductor of \(\rho\). For arbitrary \(s\) Zink made a proposal for a formula for the Swan conductor. The author gives an example where the formula does not hold. This example results from more general considerations which are of independent interest.