Let $K$ be complete local field of characteristic $0$ with ring of integers $R=\mathcal{O}_{K}$ and residue field $k=R/\mathfrak{m}_R$ of characteristic $p>0$. Let $E/K$ be an elliptic curve of split multiplicative reduction type, ie the special fibre $\mathcal{C}_k$ of its minimal proper regular model $\mathcal{C}/R$ has Kodaira type $I_n$.
This assures that we can use Tate model to study $E$. Formally this means that $E/K$ has analytic model (=apply rigidification functor) isomorphic to $\mathbb G_m / \langle q \rangle$ for appropr $q \in K^*$ of positive valuation $v_K(q) =n$.
On level of $K$-valued points of $E$ this specializes to $\phi:K^* / q^{\Bbb Z} \cong E(K)$ explicitly given by
$$ a \mapsto (X(a,q),Y(a,q)) $$
iff $a \not \in q^{\Bbb Z}$ for appropr series $X(a,q),Y(a,q)$ in $a,q$, and $\phi(a)=0$ else (see eg Silverman's ATEC, Chap. V).
Working with this Tate model unravels a lot about intrinsic geometry of the curve, resp it's integral model. Eg, if we consider the Neron model $\mathcal{E}/R (\subset \mathcal{C}/R)$ of $E$, then one can show that its special fibre $\mathcal{E}_k$ corresponds to smooth locus of special fibre of $\mathcal{C}/R)$ and consists of $n$ components each isomorphic to $\Bbb G_{m,k}$.
Then it is known that there is an iso $E(K)/E_0(K) \cong \mathcal{E}_k(k)/\mathcal{E}_k^0(k) \cong \Bbb Z/n$ "counting" irred components of the special fibre (see again Silverman's ATEC chap IV). Here $\mathcal{E}_k^0$ is identity component of special fibre of Neron model.
The pun is if we precompose this iso with Tate iso $\phi$ from bevore, the composition is given by valuation map $u \mapsto v_K(u)$ modulo valuation of $q$, and the geometric information we can immediately read up from this is in which component of special fibre $\mathcal{E}_k$ (these can be thought as enumerated by $1,2,..., v(q)=n \equiv 0$) the image of any $a \in K^* / q^{\Bbb Z} \cong E(K)$ under reduction/specialization map $\pi:E(K) \to \mathcal{E}_k(k)$.
The the point is if we start with a point identified inside the Tate model $K^* / q^{\Bbb Z}$, we can immediately see inside which component of special fibre it specializes.
The Question: If we start with explicit elliptic curve $E_q/K$ having split multipl reduction of type $I_n$ given by affine equation
$$E_q: y^2 +xy=x^3+a_4(q)x +a_6(q)$$
and we pick a point $(a,b) \in E_q(K)$, is there a an explicit formula to determine inside which component of speical fibre of Neron model its reduction lands?
This boils down keeping in mind that $K^* / q^{\Bbb Z} \to \Bbb Z/m$ is induced by valuation $v_K$, if there is an explicit formula $f((a,b)$ valued in $ \Bbb Z/n$ corresponding to $v_K \circ \phi^{-1}$?
The main "problem" so far I see is that it appears to me that it is pretty hard to write down explicitly the inverse of Tate isomorphism $\phi$, but maybe there exist explicit coordinate depending formula after composition $v_K \circ \phi^{-1}:E(K) \to \Bbb Z/n$.
The question is contextually closely related to this one I posed recently.
