I'm struggling a little bit with the definition of the connected component group of neron models. Let $E/K$ be an elliptic curve and $K$ a p-adic local field with residue class field $\kappa$. Moreover, denote by $K^{unr}$ the maximal unramified extension of $K$. I would like to know the size of the group of connected components.
What I have read so far is that I can view it as the quotient $E(K^{unr})/E^0(K^{unr})$ but I'm still not really sure about this.
And is the quotient the same as the finite etale group scheme $A_\kappa/A^0_\kappa$ over $\kappa$ where $A$ is the Neron model of $E$ over the ring of integers of $K$?
If someone could just assure me that this is right would be great, thanks!
First of all, to know the size of the group of connected components of the Néron model of an elliptic curve, you first need to know what information you have on your elliptic curve. If you have a Weierstrass equation, you can use Tate's algorithm as explained in the book by Silverman's, or just the table 4.1 of Kodaira and Néron there, which tells you its group structure.
Concretely, if the valuation of the $j$-invariant is strictly negative, then the group is either cyclic of order $-v(j)$ (if the reduction is multiplicative, called type $I_n$), or a group of order 4 (and both possibilities happen, type $I_n^*$), and if the valuation is positive, then the group can have from 0 (good reduction case) to 4 elements, depending of its type.