We study hyperelliptic curves $$y^2 = f(x)$$ 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.

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A new approach has been recently developed to study the arithmetic of hyperelliptic curves 𝑦 2 = 𝑓(𝑥) over local fields of odd residue characteristic via combinatorial data associated to the roots of 𝑓. Since its introduction, numerous papers have used this machinery of 'cluster pictures' to compute a plethora of arithmetic invariants associated to these curves. The purpose of this user's guide is to summarise and centralise all of these results in a self-contained fashion, complemented by an abundance of examples.

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We consider the K3 surfaces that arise as double covers of the elliptic modular surface of level 5, R 5,5 . Such surfaces have a natural elliptic fibration induced by the fibration on R 5,5 . Moreover, they admit several other elliptic fibrations. We describe such fibrations in terms of linear systems of curves on R 5,5 . This has a major advantage over other methods of classification of elliptic fibrations, namely, a simple algorithm that has as input equations of linear systems of curves in the projective plane yields a Weierstrass equation for each elliptic fibration. We deal in detail with the cases for which the double cover is branched over the two reducible fibers of type I 5 and for which it is branched over two smooth fibers, giving a complete list of elliptic fibrations for these two scenarios.