Collorary 3.5

Let F be a primitive cubic form and p be a prime. Then if and only if F has at least a double root modulo p, and .

Proof

If , then in particular , hence , and so by Proposition 3.4, either if F has a double root, or but then by definition of , we again have . Conversely, if F has at least a double root modulo p, and , then if it is a triple root, by definition , and if it is only a double root, by Proposition 3.4 we have , hence , hence . Note that when is at least double root modulo p, it is easily checked that the condition depends only on and modulo p.