  Let $F$ be a number field and $p\geq7$ a rational prime. We obtain a simple
descent criterion characterising those projective Galois representations
$\overline\rho:G_F\to\mathrm{PGL}_2(\mathbb{F}_p)$ for which the corresponding
twist $X_{\overline\rho}(p)$ of the principal modular curve of level $p$ is
defined over $\mathbb{Q}$. We also give a more concrete version of this result
for representations which arise from elliptic curves over cyclic number fields.
