  Classes of pairs of ternary quadratic forms parametrize quartic rings by a
result of Bhargava. We give an algorithm for finding a pair of ternary
quadratic forms that parametrize a given order of a quartic field. We examine a
new technique, essential pairs, for obtaining parameters of orders of quartic
fields from a number field database. Essential pairs for maximal orders of
quartic fields are very common and provide a simple means of obtaining an
integral basis for the ring of integers.
