Consider the elliptic curve
where I assume complex coefficients for the purposes of this question.
I am aware that there are algorithmic methods for birationally transforming a nondegenerate cubic curve into the Weierstrass canonical form (equivalently, deriving a parametrization in terms of Weierstrass elliptic functions).
I want to ask if there are analogous methods for dealing with the quartic curve given above. (Note that I've already performed polynomial depression (removing the cubic term) in advance.) In particular, I want to know if there are birational transformations that can directly convert it into the Jacobi form or some other convenient quartic standard form. (The Weierstrass form has been thoroughly covered here and in some of the answers below.)
(This is effectively a special case of this more general question.)