The Jacobians of the modular curves X ns (13) and X s (13) respectively associated with 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 X ns (13) and X s (13); this has no known "modular" explanation. An exceptional isomorphism between level 13 modular curves 931 Corollary 4.7. Let S

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... e any finite set of primes that do not divide 12. There exists an O K -linear isogeny J s (13) ∼ J ns (13) over Q whose degree is relatively prime to the primes in the set S. Proof. Since S 2 (Γ 0 (13)) = 0, we have J 0 (13) = 0. Thus, by Theorem 4.5 there is a Q-isogeny J s (13) ∼ J ns (13). In the proof of Theorem 4.5, we actually obtain this isogeny from the Z[G]-module inclusions (4.3) and (4.4) which are defined by an element in Z[G]. Since actions of GL 2 (F 13 ) and the Hecke operators away from 13 commute, this isogeny is Hecke compatible away from 13. Since the common endomorphism ring O K is generated by the Hecke operators away from 13, the isogeny is O K -linear. Residual irreducibility and the isomorphism In this section, we will use Corollary 4.7 and a study of residual Galois representations to obtain an O K -linear Q-isomorphism between J s (13) and J ns (13). As we saw at the end of Section 3, this would yield a Q-isomorphism between these Jacobians which respects canonical principal polarizations, so we could conclude via Torelli's Theorem that X ns (13) and X s (13) are isomorphic over Q. Let (5.1) θ : J s (13) → J ns (13) be an O K -linear isogeny over Q whose degree is coprime to 7 and 13 (see Theorem 5.5). We can choose such an isogeny by Corollary 4.7. Notation 5.1. In this section we denote the Jacobians J s (13) and J ns (13) by J s and J ns respectively. submodule of J s (Q). The torsion subgroups J s [I] for nonzero ideals I of O K are a natural class of finite O K [G Q ]-submodules of J s (Q). Assume ker θ = J s [I] for some I. Since K has class number 1, we have I = (α) for some nonzero α ∈ O K . Hence, θ = ψ • α for some O K -linear ψ : J s → J ns with ker ψ = 0. Thus, ψ would be a O K -linear Q-isomorphism. It remains to analyze the structure of all finite O K [G Q ]-submodules M of J s (Q) whose order is not divisible by the primes 7 and 13. Such a module M decomposes into p-primary parts for the finitely many maximal ideals p of O K in its support. Let M p be a p-primary part of M for some p ∈ Supp(M ).