For each modular curve $X_H$ of level $N$, we visualize the modular curve through the action on the $\mathrm{SL}(2, \mathbb{Z})$ fundamental domain $\mathcal{F}$ by $X_H \cap \mathrm{SL}(2, \mathbb{Z} / N\mathbb{Z})$. In order to better see distinct translations of $\mathcal{F}$, we color the fundamental domain with two separate colors and preserve these colors through translation.

Two modular curves having the same intersection with $\mathrm{SL}(2, \mathbb{Z})$ will have the same picture.

For modular curves with large index, we only show $384$ translations of $\mathcal{F}$, which slowly become transparent. The translations shown are heuristically chosen to correspond to translations with largest (Euclidean) area in the visualization.

Implementation

Pictures of modular curves were implemented by David Lowry-Duda. Details about the implementation and algorithms are contained in the technical note [Visualizing modular curves].