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Mar 18, 2016 at 12:08 comment added François Brunault Indeed $J_1(13)$ has good reduction over $\mathbf{Q}(\zeta_{13})$ (see my comment to @znt's answer). Actually over that number field $J_1(13)$ is isogenous to the product of two conjugate elliptic curves defined over $\mathbf{Q}(\sqrt{13})$. These elliptic curves have been determined in the article "Q-curves and their Manin ideals" by Josep González and Joan-C. Lario.
Mar 17, 2016 at 19:55 history answered Michael Stoll CC BY-SA 3.0