Let X be an element of finite order in GL_2(Z) such that with .
Then the characteristic polynomial of X for eigenvalues is . Since is an eigenvalue of for a positive integer k, we see that for and .
Case 1. Both and are reals. In this case, for i=1,2 are +1 or -1. Thus it has order 1 or 2.
For order 1, .
For order 2, , , .
Case 2. and are complex numbers. Since they are roots of quadratic equations with real coefficients, they are conjugate to each other. Since they are complex roots of unity, .
If , then with or for k=1,2. Thus the order of X is 4 (verify this). If , then characteristic polynomial is . In this case, X can be . It has an order 3.
If , then characterist polynomial is . In this case, X can be . It has an order 6.
Thus any element of GL2(Z) of finite order has order 1,2,3,4, or 6.