Implementation and results

The following table gives the total number of complex fields of discriminant greater or equal to to -X with for , as well as the elapsed time on a DEC Alpha.

It should be remarked that, thanks to the notion of reducedness, our algorithm gives for every cubic field a canonical equation, which we can call reduced, and which in addition has all the nice properties described in Section 4. In particular the integral basis and decomposition of primes is immediate. One consequence is that, when the cubic number field does not have a power basis, the equation that we will find will not be monic. On the other hand, if there exists a power basis, the reduced equation produced by our algorithm is not necessarily monic. We give a few examples.

• The cyclic number field with discriminant 961 can be given by the monic equation , but does not have a power basis (an integral basis is , where is a root of the above equation). Our algorithm finds the nonmonic reduced equation .
• The noncyclic number field of discriminant 1304 can be given by the monic equation , but does not have a power basis (an integral basis is ). Our algorithm finds the nonmonic reduced equation .
• The noncyclic number field of discriminant 2228 can be given by the monic equation , and this times it has a power basis. On the other hand our algorithm finds the nonmonic reduced equation .