# Degree of an isogeny in the endomorphism ring of the jacobian of a curve and self intersection index in its ring of correspondences

I hope this question is not too basic.

Let $C/\bar{k}$ be a nonsingular irreducible curve of genus $g$ and $\mathfrak{C}(C\times C)\cong \text{CH}^1(C\times C)$ be its ring of correspondences.

I am working over $k=\mathbb{F}_q$ so I am calculating the degree of a $\psi\in \text{End}_(J_C)$ (separable isogeny), what I did is to investigate it as the self intersection of the cycle corresponding to $\mathfrak{C}(C\times C)$ because both rings are in bijection, I think I got my answer partially, the problem is that I can't prove or find in the literature (Fulton) the relation between the degree of an endomorphism of the jacobian and its corresponding $1$-cycle self intersection index in the ring of correspondences. Thank you.