  Two birational projective varieties in $P^n$ are Cremona Equivalent if there
is a birational modification of $P^n$ mapping one onto the other. The minimal
Cremona degree of $X\subset P^n$ is the minimal integer among all degrees of
varieties that are Cremona Equivalent to $X$. The Cremona Equivalence and the
minimal Cremona degree is well understood for subvarieties of codimension at
least $2$ while both are in general very subtle questions for divisors. In this
note I compute the minimal Cremona degree of quartic surfaces in $P^3$. This
allows me to show that any quartic surface of elliptic ruled type has non
trivial stabilizers in the Cremona group.
