  Using Mazur's theorem on torsions of elliptic curves, an upper bound 24 for
the order of the finite Galois group $\mathcal{H}$ associated with weighted
walks in the quarter plane $\mathbb{Z}^2_+$ is obtained. The explicit criterion
for $\mathcal{H}$ to have order 4 or 6 is rederived by simple geometric
argument. Using division polynomials, a recursive criterion for $\mathcal{H}$
having order $4m$ or $4m+2$ is also obtained. As a corollary, explicit
criterion for $\mathcal{H}$ to have order 8 is given and is much simpler than
the existing method.
