Definition 6.1 Let be an integral binary complex cubic form, and let as above. We say that F is reduced if 0<|Q|<P<R and if in addition a>0, , and d>0 if b=0.
Note that when F is irreducible, is irrational, hence the special cases Q=0, P=|Q| or P=R which occurred in the real case cannot occur here. Another nice fact is that we do not need to compute the irrational numbers P, Q and R at all:
Let be a complex cubic form. Then F is reduced if and only if:
For this notion to be useful, we must have the analogue of Proposition 5.4.
- Two equivalent reduced complex cubic forms are equal.
- A reduced complex cubic form belonging to U is irreducible.
- Any irreducible complex cubic form is equivalent to a unique reduced form.
We have bounds on the coefficients of a reduced complex cubic form as follows.
Let be a reduced form such that . We have the following inequalities:
In the actual implementation we will proceed essentially as in the real case. The analogue of Theorem 5.6 is as follows.
Let (resp. ) be the number of equivalent complex cubic forms (resp. of isomorphism classes of complex cubic fields) of discriminant greater than or equal to -X. Then as , we have: for any .
Once again, we see that the algorithm will be linear in X, and the number of loops will be approximately times the number of cubic fields found.