An exceptional isomorphism between modular curves of level 13

Abstract

In this note we use representation theory to compute an explicit equation over Q for the modular curve Xns(13) associated to the normalizer of a non-split Cartan subgroup of level 13. We also prove that the same equation defines the modular curve Xs(13) associated to the normalizer of a split Cartan subgroup of level 13. Hence, these two modular curves are isomorphic over Q. This isomorphism does not seem to have a “modular” explanation. For instance, Xs(13)(Q) contains a cusp but Xns(13)(Q) does not.