The Hessian of a genus one curve

Abstract

We continue our development of the invariant theory of genus one curves with the aim of computing certain twists of the universal family of elliptic curves parametrised by the modular curve X(n) for n = 2,3,4,5. Our construction makes use of a covariant we call the Hessian, generalising the classical Hessian that exists in degrees 2 and 3. In particular we give explicit formulae and algorithms for computing the Hessian in degrees 4 and 5. This leads to a practical algorithm for computing equations for visible elements of order n in the Tate-Shafarevich group of an elliptic curve. Taking Jacobians we also recover the formulae of Rubin and Silverberg for families of n-congruent elliptic curves.

42 pages; section on visibility added

References in corpus (3)

• Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves
• Explicit n-descent on elliptic curves, I. Algebra
• Explicit n-descent on elliptic curves. III. Algorithms

Cited by in corpus (11)

• Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves
• The arithmetic of genus two curves with (4,4)-split Jacobians
• Explicit n-descent on elliptic curves. III. Algorithms
• On families of n-congruent elliptic curves
• Invariant theory for the elliptic normal quintic, II. The covering map
• Descent on elliptic curves
• Hesse Pencils and 3-Torsion Structures
• Explicit Families Of Elliptic Curves With The Same Mod 6 Representation
• Minimisation and reduction of 5-coverings of elliptic curves
• Visibility of 4-covers of elliptic curves
• Congruences of elliptic curves arising from non-surjective mod $N$ Galois representations