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The binary quartic is only birationally equivalent to a Weierstrass equation if it has a rational point. Then, the required transformations are indeed to be found in Cassels. They have not yet been implemented in Sage. If there is no rational point (or you do not know one) then there is a degree 4 map from the quartic to its Jacobian elliptic curve, called a two-covering map. These are used in two-descent, but there is no Sage function which simply inputs a quartic and outputs the cubic. Note that getting the equation of the Jacobian cubic is easy: it is Y^2=X^3-27IX-27*J where I and J are the classical invariants of the quartic. You can read more about this in my paper Classical Invariants and 2-descent on elliptic curves (preprint here), Journal of Symbolic Computation (Proceedings of the Second Magma Conference, Milwaukee, May 1996), Jan/Feb 2001, pages 71-87.