#### Vol. 4, No. 6, 2010

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Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves

### John E. Cremona, Tom A. Fisher and Michael Stoll

Vol. 4 (2010), No. 6, 763–820
##### Abstract

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 $ℚ$, we develop a theory of reduction and again give explicit algorithms for $n=2$, $3$ and $4$.

##### Keywords
elliptic curves, genus-one curves, minimisation, reduction, descent
##### Mathematical Subject Classification 2000
Primary: 11G05
Secondary: 11G07, 11G05, 14H52, 14H25