It was first pointed out by Weil that we can use classical invariant theory to compute the Jacobian of a genus one curve. The invariants required for curves of degree n = 2,3,4 were already known to the nineteenth centuary invariant theorists. We have succeeded in extending these methods to curves of degree n = 5, where although the invariants are too large to write down as explicit polynomials, we have found a practical algorithm for evaluating them.
This paper has appeared in the Proceedings of the London Mathematical Society. The formulae and algorithms in Sections 7 and 8 have been implemented in Magma Version 2.13 by the author. The code referred to in Section 9 is available as two files here and here. A much earlier version of the paper, and a pari implementation are available here.