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Minimal Models for 2-coverings of Elliptic Curves

Published online by Cambridge University Press:  01 February 2010

Michael Stoll  and
John E. Cremona
Michael Stoll
Affiliation:
School of Engineering and Science, International University Bremen, P.O.Box 750561, 28725 Bremen, Germany, m.stoll@iu-bremen.de, http://www.iu-bremen.de/directory/faculty/29179/
John E. Cremona
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, john.cremona@nottingham.ac.uk, http://www.maths.nottingham.ac.uk/personal/jec/

Abstract

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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.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 5 , 2002 , pp. 220 - 243
Copyright
Copyright © London Mathematical Society 2002

References

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