Minimal Twists of modular curves (awaiting review)

Open subgroups $H_1,H_2\le \GL_2(\widehat \Z)$ for which $H_1\cap \SL_2(\widehat\Z) = H_2\cap SL_2(\widehat \Z)$ are said to be twists. The corresponding modular curves $X_{H_1}$ and $X_{H_2}$ become isomorphic after base change to any cyclotomic field $\Q(\zeta_n)$ where $n$ is divisible by the level of both $H_1$ and $H_1$.

The minimal twist of a modular curve $X_H$ with $\det(H)=\widehat \Z^\times$ is the modular curve whose label is minimal among all twists with $\det(H)=\widehat\Z^\times$ (in particular, it has minimal level).