We give an algorithm to compute the conductor for curves of genus 2. It is based on the analysis of 3-torsion of the Jacobian for genus 2 curves over 2-adic fields.

We present a family of algorithms for computing the Galois group of a polynomial defined over a [Formula: see text]-adic field. Apart from the “naive” algorithm, these are the first general algorithms for this task. As an application, we compute the Galois groups of all totally ramified extensions of [Formula: see text] of degrees 18, 20 and 22, tables of which are available online.

We give a brief re-exposition of the theory due to Pauli and Sinclair of ramification polygons of Eisenstein polynomials over p-adic fields, their associated residual polynomials and an algorithm to produce all extensions for a given ramification polygon. We supplement this with an algorithm to produce all ramification polygons of a given degree, and hence we can produce all totally ramified extensions of a given degree.

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