here is my question :
Let $K$ be any field, $ E \to spec(K)$. Let $ v \in M_K $ an archimedean place. 
We know that $ \overline{K_v} \simeq \mathbb{C}$ and there exists $\tau_v \in \mathbb{H}$ such that :
$$ E(\overline{K_v}) \simeq \mathbb{C} / \mathbb{Z} + \tau_v \mathbb{Z} $$
My question is : can we have an upper bound for $Im(\tau_v)$ for each $v$ ?