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In

The place to look for this is Chapter 4 ("The Neron Model") of Silverman's book Advanced Topics in the Arithmetic of Elliptic Curves, you will find the following fundamental theorem (and its proof!)specifically Theorem 4.6.1: the identity component of 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 $N$.

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In Chapter 4 of Silverman's book Advanced Topics in the Arithmetic of Elliptic Curves, you will find the following fundamental theorem (and its proof!): the identity component of 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 $N$.