
A genus one curve of degree 5 is defined by the 4 × 4 Pfaffians of a 5 × 5 alternating matrix of linear forms on P^{4}. We describe a general method for investigating the invariant theory of such models. We use it to explain how we found our algorithm for computing the invariants [12] and to extend our method in [14] for computing equations for visible elements of order 5 in the TateShafarevich group of an elliptic curve. As a special case of the latter we find a formula for the family of elliptic curves 5congruent to a given elliptic curve in the case the 5congruence does not respect the Weil pairing. We also give an algorithm for doubling elements in the 5Selmer group of an elliptic curve, and make a conjecture about the matrices representing the invariant differential on a genus one normal curve of arbitrary degree.