Let $E$ be an elliptic curve over $\mathbf{Q}$ which has split multiplicative reduction at $p$ (a prime). If one chooses a global Neron model of $E$ over $\mathbf{Z}$ (unique up to unique isomorphism over $\mathbf{Z}$) then one gets a holomorphic differential (on the Neron model) which is well defined up to $\pm 1$. Then one may define the following positive real number $$ \Omega_E:=\int_{E(\mathbf{R})}|\omega| \in \mathbf{R}_{>0}. $$

The quantity $\Omega_E$ "seems to appear" in the $p$-adic formulation of BSD. Let $\alpha,\alpha'\in\mathbf{C}$ be the two roots of the Frobenius at $p$ (acting on the Tate module). Let us assume that $\alpha$ is ordinary i.e. that its image in $\mathbf{C}_p$ is a unit ($v_p(\alpha)=0$). Note here that the choice $\alpha$ (the ordinary root at $p$) depends (in general) on the choice of an embedding $\bar{\mathbf{Q}}\rightarrow \mathbf{C}_p$. Once such an embedding is fixed then the $p$-adic $L$-function $L_p(E,s)$ (which is defined by $p$-adic integration using a certain measure which depends on $\alpha$), seems to be, as far as I understand, completely determined (may be this is here that I make a mistake). Therefore "the image" of $\Omega_E$ in $\mathbf{C}_p$ should also be completely determined. But if $\Omega_E$ is transcendental over $\mathbf{Q}$ then one can send $\Omega_E$ on any transcendental (over $\mathbf{Q}$) element of $\mathbf{C}_p$ by choosing a suitable embedding of $\mathbf{R}\hookrightarrow \mathbf{C}_p$.

Q: So how can we reconcile the fact that $\Omega_E$ admits many different embeddings inscide $\mathbf{C}_p$ with the observation that its image in $\mathbf{C}_p$ should be completely determined?

*added*: Is this discrepancy somehow compensated by the presence of the $\mathcal{L}$ invariant...?