Let $C$ be a $\mathbb{Q}$-curve with no complex multiplication. In this note we characterize the number fields $K$ such that there is a curve $C^{\text{'}}$ isogenous to $C$ having all the isogenies between its Galois conjugates defined over $K$, and also the curves $C^{\text{'}}$ isogenous to $C$ defined over a number field $K$ such that the abelian variety Res${}_{K/\mathbb{Q}}(C^{\text{'}}/K)$ obtained by restriction of scalars is a product of abelian varieties of GL${}_2$-type.

We describe a process for defining and computing a fundamental domain in the upper half plane $\mathscr{H}$ of a Shimura curve $X_0^D\left(N\right)$ associated with an order in a quaternion algebra $A/𝐐$. A fundamental domain for $X_0^D\left(N\right)$ realizes a finite presentation of the quaternion unit group, modulo units of its center. We give explicit examples of domains for the curves $X_0^6\left(1\right),X_0^{15}\left(1\right),\text{and}{X}_0^{35}\left(1\right)$. The first example is a classical example of a triangle group and the second is a corrected version of that appearing in the book of Vignéras [13], due to Michon....