Is there a way to find the number of solutions of equations in this form $$x^{11}+12x^{7}+2x^{2}+10 \equiv 0 \text{ (mod 15)}$$ in $\mathbb{Z}_{15}$ or more general $\text{(mod n)}$ in $\mathbb{Z}_{n}$?
I know that I can write the equation as a set of equations $$\text{ (mod 3)}$$ and $$\text{ (mod 5)}$$ or in general the solutions are given using the Chinese reminder theorem solving the set of equations $$\text{ (mod pq...v)}$$ where p and q are prime factors of n. I would like to know if there is a specific method to find the number of solutions without computing them with the Chinese reminder theorem.