joint with John Cremona
We give an extension and correction to a result stated in the first author's paper "Classical Invariants and 2-descent on elliptic curves", J. Symb. Comp. 31 (2001), 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.