I'm currently facing the problem of computing chern classes for Varieties. More precisely the product of such varieties.

Let $C_i$ be a variety in $\mathbb{CP}^2$ given by the Weierstraß $\wp$-map. I want to construct a product of 3 such varieties. Nothing fancy, just $C_1\times C_2 \times C_3$ and calculate it's chern classes.

I'm specifically interested in the second chern class of the Tangent Bundle and some Vector bundle. However, I'm having real trouble actually starting some calculation.

In the case of a single variety in $\mathbb{CP}^2$ I could have used the splitting principle and used the fact that the Normal bundle is a Line Bundle to calculate the total chern class.

However in the case of 3 such varieties, the first problem that arises is, I don't even know where it lies in? According to Segre Embedding I'd say $\mathbb{CP}^{26}$, but that seems a bit high. Perhaps $\mathbb{CP}^4$ would suffice? $\mathbb{CP}^{2+2+2}$? However, this would only help me with the Tangent Bundle.

Could anyone give me some pointers on how to calculate the total chern class in such a case or some reference where it is done in a similar case? Thanks!