Invariants of the elliptic normal quintic

Let k be a field with char k 6 = 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