Let $K$ be a local field that is complete with respect to a discrete valuation. When an elliptic curve, $E/K$, has reduction type represented by the Kodaira symbol $I_{2n}^{*}$, its component group can be ${\mathbb Z}/2{\mathbb Z} \times {\mathbb Z}/2{\mathbb Z}$.

In this case, is there a criterion to distinguish a point on the components $(1,0)$ and $(0,1)$ from a point on the component $(1,1)$?

In Lemma 5.1 of Silverman's ``Computing Heights on Elliptic Curves'', Math Comp. v. 51 (1988), 339--358, he shows how to determine the component when we have split multiplicative reduction. Ideally something like this is the sort of thing I am looking for.

Let $K$ be a local field that is complete with respect to a discrete valuation. When an elliptic curve, $E/K$, has reduction type represented by the Kodaira symbol $I_{2n}^{*}$, its component group can be ${\mathbb Z}/2{\mathbb Z} \times {\mathbb Z}/2{\mathbb Z}$.

In this case, is there a criterion to determine which component a point is on? In particular, to tell if it is on the component $(1,0)$ (or $(0,1)$) or on the component $(1,1)$?

In Lemma 5.1 of Silverman's ``Computing Heights on Elliptic Curves'', Math Comp. v. 51 (1988), 339--358, he shows how to determine the component when we have split multiplicative reduction. Ideally something like this is the sort of thing I am looking for.