Cremona, John E.; Fisher, Tom A.; Stoll, Michael Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves. (English) Zbl 1222.11073 Algebra Number Theory 4, No. 6, 763-820 (2010). Summary: We consider models for genus-one curves of degree \(n\) for \(n = 2\), 3 and 4, which arise in explicit \(n\)-descent on elliptic curves. We prove theorems on the existence of minimal models with the same invariants as the minimal model of the Jacobian elliptic curve and provide simple algorithms for minimising a given model, valid over general number fields. Finally, for genus-one models defined over \(\mathbb Q\), we develop a theory of reduction and again give explicit algorithms for \(n = 2\), 3 and 4. Cited in 1 ReviewCited in 26 Documents MSC: 11G05 Elliptic curves over global fields 11G07 Elliptic curves over local fields 14H52 Elliptic curves 14H25 Arithmetic ground fields for curves Keywords:elliptic curves; genus-one curves; minimisation; reduction; descent PDFBibTeX XMLCite \textit{J. E. Cremona} et al., Algebra Number Theory 4, No. 6, 763--820 (2010; Zbl 1222.11073) Full Text: DOI arXiv