I believe there are only finitely many congruence subgroups $\Gamma\le SL_2(\mathbb{Z})$ such that the compactification of $\mathcal{H}/\Gamma$ is genus 1.

Is there somewhere I can find a list of these genus 1 modular curves and look at their $j$-invariants (and ideally also reduction types over their field of definition). Here, I'm referring to the moduli-theoretic model of the elliptic curve, which is necessarily a quotient of some $X(n)$ (which is defined over $\mathbb{Q}(\zeta_n)$) and hence the $j$-invariant will lie in $\mathbb{Q}(\zeta_n)$.

In particular I'm interested in whether or not you can say anything about the integrality of the $j$-invariants of these genus 1 modular curves, especially in the case when the congruence subgroup is torsion-free. For example, are they all integral? are they all non-integral? Do they all have additive/multiplicative reduction? Do they have CM?