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

Abstract

In this paper 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.

56 pages

References in corpus (4)

• Explicit n-descent on elliptic curves, I. Algebra
• The Hessian of a genus one curve
• The invariants of a genus one curve
• Explicit n-descent on elliptic curves, II. Geometry

Cited by in corpus (18)

• The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point
• The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1
• The Hessian of a genus one curve
• Explicit n-descent on elliptic curves. III. Algorithms
• On the number of cubic orders of bounded discriminant having automorphism group $C_3$, and related problems
• Explicit n-descent on elliptic curves, II. Geometry
• On families of n-congruent elliptic curves
• Second p descents on elliptic curves
• Minimal Genus One Curves
• Minimization of hypersurfaces
• Computing minimal Weierstrass equations
• The proportion of plane cubic curves over $\mathbb{Q}$ that everywhere locally have a point
• Counting Models of Genus One Curves
• The second moment of the number of integral points on elliptic curves is bounded
• Invariants of models of genus one curves and modular forms
• Minimisation and reduction of 5-coverings of elliptic curves
• Visibility of 4-covers of elliptic curves
• Rational points on curves