Minimal Models for 2-coverings of Elliptic Curves

This paper concerns the existence and algorithmic determination of minimal models for curves of genus 1, given by equations of the form y = Q(x) where Q(x) has degree 4. These models are used in the method of 2-descent for computing the rank of an elliptic curve. Our results are complete for unramified extensions ofQ2 and Q3 and for all p-adic fields for p > 5. Our primary motivation is to complete the results of Birch and Swinnerton-Dyer [2], which are incomplete in the case of Q2. Our results in this case (when applied to 2-coverings of elliptic curves over Q) yield substantial improvements in the running times of the 2-descent algorithm implemented in the program mwrank [5]. The paper ends with a section on implementation and examples, and an appendix gives constructive proofs in sufficient detail to be used for implementation.