The Hessian of a genus one curve

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.