### 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**.