Published online by Cambridge University Press: 01 February 2010
This paper concerns the existence and algorithmic determination of minimal models for curves of genus 1, given by equations of the form y2 = 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. The results described here are complete for unramified extensions of Q2 and Q3, and for all p-adic fields for p greater than or equal to 5. The primary motivation for this work was to complete the results of Birch and Swinnerton-Dyer, which are incomplete in the case of Q2. The 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. The paper ends with a section on implementation and examples, and an appendix gives constructive proofs in sufficient detail to be used for implementation.