  The amplitude for the singlet channels in the 4-point function of the
fundamental field in the conformal field theory of the 2d $O(n)$ model is
studied as a function of $n$. For a generic value of $n$, the 4-point function
has infinitely many amplitudes, whose landscape can be very spiky as the higher
amplitude changes its sign many times at the simple poles, which generalize the
unique pole of the energy operator amplitude at $n=0$. In the stadard
parameterization of $n$ by angle in unit of $\pi$, we find that the zeros and
poles happen at the rational angles, forming a hierarchical tree structure
inherent in the Poincar\'{e} disk. Some relation between the amplitude and the
Farey path, a piecewise geodesic that visits these zeros and poles, is
suggested. In this hierarchy, the symmetry of the congruence subgroup
$\Gamma(2)$ of $SL(2,\mathbb{Z})$ naturally arises from the two clearly
distinct even/odd classes of the rational angles, in which one respectively
gets the truncated operator algebras and the logarithmic 4-point functions.
