Does $\phi(a)=n!$ have finitely many solutions in integers?

Question

I tried to treat solution of $\phi(a)=n!$ in integers such that $\phi(n)$ is the Euler totient function and $a$ is an integer, for $a=n$ it's clear that there is no solution because $\phi(n)<n<n!$. Some solutions I got by wolfram alpha are mentioned here. For instance, we have $(a,n)=(9,3)$ and $(-9,3)$, $(7,3)$ and $(-7,3)$ ,$(6,2)$ and $(-6,2)$, $(4,2)$ and $(-4,2)$,$(3,2)$ and $(-3,2)$, $(2,0)$ and $(-2,0),(2,1)$ and $(-2,1)$, $(1,0)$ and $(-1,0)$. I suspect there are finitely many of them, but this needs a proof. My question here is: When does $\phi(a)=n!$ have a solution in integers? Does it have a finite number of solutions?