What is the Frattini subgroup of $SL(2,\mathbb{Z}_p)$? If $\Phi$ denotes this Frattini subgroup and $\Gamma(p^r)$ denotes the kernel of the projection to $SL(2,\mathbb{Z}/p^r)$, then since $\Gamma(p)$ is pro-$p$, we have $\Phi\ge\Gamma(p^2)$.
However I believe $\Phi = \Gamma(p)$. Is this true? If possible I'd like to see an abstract group-theoretic argument, as well as a more geometric $p$-adic Lie-group style argument, if they exist.
Edit: Let's assume $p\ge 5$ for simplicity.