The place to look for this is Chapter 4 ("The Neron Model") of Silverman's book Advanced Topics in the Arithmetic of Elliptic Curves, specifically Theorem 4.6.1: the Neron model of an elliptic curve is obtained by removing the singular points from the minimal regular proper model.
Thus in your case the connected component is a rational curve with two points removed: as a group it is $\mathbb{G}_m$, the multiplicative group. The component group here is cyclic of order $N$.