Contains $-I$ (awaiting review)

A modular curve $X_H$ with $-I\in H\in \GL_2(\widehat\Z)$ has the property that if an elliptic curve $E$ over a number field $K$ has $j$-invariant $j(E)\ne 0,1728$ in the image of $j\colon X_H\to X(1)$, then the mod-$N$ Galois image of every twist of $E$ is conjugate to a subgroup of the projection of $H$ to $\GL_2(\Z/N\Z)$. Modular curves $X_H$ with $-I\not\in H$ do not enjoy this property.