Let $K$ be a finite extension of $\mathbb{Q_p}$. The group $\ker H^1(G_K, \mathbb{Q}_p) \rightarrow H^1(G_K, B_{crys})$ is one-dimensional, which tells us that among all extensions of Galois modules

$$ 0 \rightarrow \mathbb{Q}_p \rightarrow E \rightarrow \mathbb{Q}_p \rightarrow 0, $$

which are classified by $H^1(G_K, \mathbb{Q}_p) \cong Hom_{cts}(K^*, \mathbb{Q}_p)$, there is a one-dimensional subspace corresponding to *crystalline* representations.

My question is basically: What homomorphism $K^* \rightarrow \mathbb{Q}_p$ does this crystalline cocycle correspond to?

Actually, I know the answer. It was explained to me that a theorem of Sen asserts that a crystalline representation with only $0$ as Hodge-Tate weight must have finite image of inertia, which in this case must be trivial, i.e. unramified. So the homomorphism in question is (a scalar multiple of) $x \mapsto val_p(x)$.

I would like to come to a more elementary and concrete understanding of this puzzle. The fact that this is a coboundary in $H^1(G_K, B_{crys})$ means that I can find $b \in B_{crys}$ such that inertia acts trivially on $b$ and Frobenius translates $b$ by $1$. Is it possible to describe this period explicitly in terms of the construction of $B_{crys}$?

As someone who has never really understood the definition of $B_{crys}$, it would be even more satisfying to me to have an explicit description in terms of elements which I believe should belong in $B_{crys}$ if my intuitive definition of the latter is merely ``the ring of periods for cohomology of varieties with good reduction''. In other words, I would love to see this period $b$ expressed in terms of periods coming from cohomology of familiar varieties, such as the period for the cyclotomic character.