  We study hyperelliptic curves y^2=f(x) over local fields of odd residue
characteristic. We introduce the notion of a "cluster picture" associated to
the curve, that describes the p-adic distances between the roots of f(x), and
show that this elementary combinatorial object encodes the curve's Galois
representation, conductor, whether the curve is semistable, and if so, the
special fibre of its minimal regular model, the discriminant of its minimal
Weierstrass equation and other invariants.
