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We now consider the case where , * i.e * when **F** corresponds to
a complex cubic field. In this case the Hessian is an indefinite quadratic
form, and in general there will be many reduced quadratic forms equivalent
to it. Instead of using the Hessian, we will use a clever idea due to Matthews
and Berwick (see [9]). If , then **F** has a unique real root ,
and if **F** is irreducible, so if we factor **F** in as
the quadratic form will be definite (* i.e * ), but with
real nonrational coefficients. We are going to show that the form
has many of the properties of the Hessian.

An easy computation gives
By changing to , * i.e * **F** into **-F**, we may assume that
, and if we have
hence
If , then a simple computation
shows that
the absolute value sign coming from the choice .
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:

### Lemma 6.2

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.

### Proposition 6.3

- 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.

### Lemma 6.4

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.

### Theorem 6.5

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.

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