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Complex cubic fields

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:

[Proof]

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.

[Proof]

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:

[Proof]

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.

Contents Next: Implementation and results Up: Binary Cubic Forms and Previous: Remark