$\DeclareMathOperator\PSL{PSL}$Let $ \mathbb{Q} $ be the field of rational numbers and $ \mathbb{R} $ the field of real numbers. Let $ \Sigma_g $ be a surface of genus $ g \geq 2 $. Let $ \pi_1(\Sigma_g) $ be the fundamental group of the surface. There are many way to embed $ \pi_1(\Sigma_g) $ into $\PSL_2(\mathbb{R}) $. There are, however, no ways to embed $ \pi_1(\Sigma_g) $ into $ \PSL_2(\mathbb{Q}) $. Given some $ g \geq 2 $, is there a good way (an algorithm) to find an intermediate field $ \mathbb{F} $ between $ \mathbb{R} $ and $ \mathbb{Q} $ (preferably of minimal degree over $ \mathbb{Q} $) such that $ \pi_1(\Sigma_g) $ embeds in $ \PSL_2(\mathbb{F}) $?