Theorem 4.7

Let be a cubic form. This algorithm outputs true or false according to whether F corresponds to the image of a cubic field by the Davenport-Heilbronn map.

• If F is not irreducible, return false.
• If F is not primitive, return false.
• Compute the Hessian by the formulas given. Set , (so ). Using Proposition 3.3, check that and , otherwise return false.
• If for some p>3, return false.
• Set . Remove all powers of 2 and 3 from t (in fact at most and ), and let again t be the result. If return false.
• If t is squarefree return true, otherwise return false.