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