On the equivalence of binary quartics

We give an extension and correction to a result stated in the first author’s paper [Cremona, J.E., 2001. Classical invariants and 2-descent on elliptic curves. J. Symbolic Comput. 31 (1–2), 71–87; Computational algebra and number theory. Milwaukee, WI, 1996], concerning the equivalence of binary quartics. In the earlier version the cases where I=0 or J=0 were not fully treated, and neither were the cases of reducible quartics or those whose resolvent cubic is reducible; these are dealt with here. We also give an alternative criterion for equivalence.