Let k be a field of characteristic not equal to 2,3,5. Let C be a smooth curve of genus one defined over k. Suppose that C admits a k-rational divisor of degree n. For n = 2,3,4 classical invariant theory allows us to compute the Jacobian of C without making any field extensions. We extend to the case n = 5 where although the corresponding invariants are too large to write down as polynomials, we have found a practical algorithm for computing them.
We have implemented our algorithm in the case k = Q, using pari / gp. Our program is called jacalg5.gp. This article has been superceded by the version available here.