Transcendence of the Hodge-Tate filtration

For $C$ a complete algebraically closed extension of $\mathbb{Q}_p$, we show that a one-dimensional $p$-divisible group $G/ \mathcal{O}_C$ can be defined over a complete discretely valued subfield $L \subset C$ with Hodge-Tate period ratios contained in $L$ if and only if $G$ has CM, if and only if the period ratios generate an extension of $\mathbb{Q}_p$ of degree equal to the height of the connected part of $G$. This is a $p$-adic analog of a classical transcendence result of Schneider which states that for $\tau$ in the complex upper half plane, $\tau$ and $j(\tau)$ are simultaneously algebraic over $\mathbb{Q}$ if and only if $\tau$ is contained in a quadratic extension of $\mathbb{Q}$. We also briefly discuss a conjectural generalization to shtukas with one paw.