Consider the following setting. I have a map from the Legendre family $Y$ to $\mathbb{P}_1\setminus \{0,1,\infty\}(\mathbb{Z})$. Call this map $f$, and I am trying to understand the monodromy representation of $\pi_1$ over $(R^1f_*\mathbb{Q}_p)_x$, where $\pi_1$ stands for the etale fundamental group, and $x$ is a point on $\mathbb{P}_1\setminus \{0,1,\infty\}(\mathbb{Z})$.
I know that $(R^1f_*\mathbb{Q}_p)_x$ is isomorphic to a direct sum of two copies of $\mathbb{Q}_p$, and that the monodromy representation, denoted $\rho$, has image contained in $SL_2(\mathbb{Z}_p)$, but I would like to know what it is precisely. Also, I would like to know what is the image of the geometric fundamental group under $\rho$, and since the absolute Galois group of $\mathbb{Q}$ naturally embeds inside the etale fundamental group, I would also like to know what is its image under $\rho$, and specifically what is the image of the Frobenius at $p$, if that makes sense.
Thanks in advance!