An exceptional isomorphism between level 13 modular curves via Torelli’s Theorem

Abstract

The Jacobians of the modular curves Xns(13) and Xs(13) respectively associated to the normalizers of non-split and split Cartan subgroups of level 13 are isogenous over Q. In this note, we construct a Q-isomorphism between these Jacobians which respects their canonical principal polarizations. In particular we obtain a Q-isomorphism between Xns(13) and Xs(13); this has no known \modular explanation.