
joint with John Cremona and Michael Stoll
In this paper we consider models for genus one curves of degree n for n = 2, 3 and 4, which arise in explicit ndescent 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.