An exceptional isomorphism between modular curves of level 13 (Q741683)

Let \(X_{\mathrm{s}}(13)\) (resp. \(X_{\mathrm{ns}}(13)\)) be the modular curve associated to the normalizer of a split Cartan group (resp. non-split Cartan group) of level \(13\). The author shows that those curves are defined by the same equation over \(\mathbb Q\) by constructing a basis of their spaces of regular \(1\)-forms from the cuspidal representations of \(\mathrm{GL}_2(\mathbb{F}_{13})\) and finds seven \(\mathbb Q\)-rational points on each of them, which except one rational cusp on \(X_{\mathrm{s}}(13)\) give all \(13\) CM-points of class number \(1\). The author mentions that this isomorphism does not seem to have a ``modular'' explanation. Further the author computes explicitly the \(j\)-line map of those curves to the modular curve associated to \(\mathrm{SL}_2(\mathbb Z)\).