• Step 1 will in practice not be necessary since we will always use this algorithm with reduced forms (see later), which are automatically irreducible when they belong to U.
• Although Step 2 seems to be necessary, this is in fact not the case since nonprimitive forms will be excluded in the subsequent steps. Indeed, let p be a prime dividing all the coefficients of F. Then clearly . If p>3, Step 4 will return false. Assume now p=2 or p=3. Then and F has at least a double root modulo p. If F has only a double and not a triple root, then . On the other hand if F has a triple root, i.e if then with and hence so and are equal to zero modulo p, hence .
• Even though it seems useful to include the unnecessary Step 2, it can be shown that on average it slows the algorithm, so in the final form thanks to the preceding remark we will suppress it.

We end this section by the following proposition.

Proposition 4.8

Let K be a cubic number field, and as above write where is a fundamental discriminant. Let be the cubic form associated to K by the Davenport-Heilbronn map, and let its Hessian. Finally, set . Then:

• or and apart from powers of 3, f and are squarefree.
• If then .
• If then .
• In particular , f and have the same prime divisors, and we can have only equal to , , and .