Computation of the class number and class group of a complex cubic field

Let h and G be, respectively, the class number and the class group of a complex cubic field of discriminant A. A method is described which makes use of recent ideas of Lenstra and Schoof to develop fast algorithms for finding h and G. Under certain Riemann hypotheses it is shown that these algorithms will compute h in O(1A 1/5 ? +) elementary operations and G in O(1A 1/4 +?) elementary operations. Finally, the results of running some computer programs to determine h and G for all pure cubic fields.9(VDi), with 2 S D < 30,000, are summarized.