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A genus one curve of degree 5 is defined by the \(4 \times 4\) Pfaffians of a \(5 \times 5\) alternating matrix of linear forms on \(\mathbb{P }^4\). 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 and to extend our method for computing equations for visible elements of order 5 in the Tate-Shafarevich group of an elliptic curve. As a special case of the latter we find a formula for the family of elliptic curves 5-congruent to a given elliptic curve in the case the 5-congruence does not respect the Weil pairing. We also give an algorithm for doubling elements in the \(5\)-Selmer 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.

Invariant theory for the elliptic normal quintic, I. Twists of X(5)