Very roughly, my question is: given a geodesic in the modular curve $X_0(11)$, how do you find its homology class?
In more detail: a convenient way to specify a geodesic in $X_0(11)$ is to first specify a geodesic in the completed upper-half plane $\mathbf{H}^* = \mathbf{H} \cup \mathbf{P}^1(\mathbf{Q})$ and then push it forward to $X_0(11)$ under the natural map $$\mathbf{H}^* \to X_0(11).$$ In my case, I'm looking at the geodesic in $\mathbf{H}^*$ connecting the rational number $\frac{1}{11} \in \mathbf{P}^1(\mathbf{Q})$ to the point $\infty \in \mathbf{P}^1(\mathbf{Q})$ in the completed upper half plane $\mathbf{H}^*$. (This geodesic is just a vertical line connecting $\frac{1}{11}$ to $\infty$ in the upper half plane.)
The points $\frac{1}{11}$ and $\infty$ are $\Gamma_0(11)$-equivalent so they are sent to the same point of $X_0(11)$ under the natural map $\mathbf{H}^* \to X_0(11)$. Therefore, the geodesic connecting them becomes a loop in $X_0(11)$. I want to know: what is the homology class of this loop? That is, which "hole" of the torus $X_0(11)$ does it surround, and how many times does it go around that hole?
I'm quite stuck as to how to answer this. It seems to me that we only have information about the start and end points of the geodesic, and there's some strange topological weirdness going on in the middle that I don't know how to get my hands around. I've tried looking at the image of the geodesic in a fundamental domain for $X_0(11)$, but that hasn't been too successful either. Any help would be greatly appreciated. :) Thank you very much, friends!
EDIT: I've been asked to provide a picture of the fundamental domain of $X_0(11)$, so here it is:
The fundamental domain is the area above the green and blue circles on the bottom, and contained within the red lines on either side. The sides of the same colors are identified. If you make these identifications, you'll get a torus with two points missing, which corresponds to the fact that $X_0(11)$ has two cusps. Also, the circles have diameters $1/3, 1/6, 1/6,$ and $1/3$ respectively from left to right, totalling to a length of $1$.
The difficulty is, the geodesic I'm interested in isn't contained in the fundamental domain so I can't see what its image in the torus would be. Furthermore, I can't seem to figure out what its image in the fundamental domain would be. (This is because there any infinitely many matrices in $\Gamma_0(11)$, it is not clear how to know which matrix brings the geodesic connecting $\frac{1}{11}$ to $\infty$ into the fundamental domain.)