# Self intersection of correspondences of a curve $X$ and degree of isogenies in $\text{End}(J_X)$

I migrated my question from MO because I just got one vote and maybe was 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 have two questions:

a) 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.

b) As mentioned, there's an isomorphism $\Psi:\mathfrak{C}(C\times C)\to \text{End}(J_C)$ and if $X$ is a non trivial correspondence (not a fiber of a projection to $C\times C$) then $\Psi(X)$ and $\Psi(X^t)$ are endomorphisms where $t$ denotes the transpose of the correspondence (interchange of coordinates in $C\times C$), it is known that $\Psi(X^t)$ is the Rosati involution on of $\Psi(X)$ on $\text{End}(J_C)$. To answer indirectly my first question, I want to know how can I calculate for $X\in \mathfrak{C}(C\times C)$ the correspondence $X^*$ such that $\Psi(X^{*})\circ \Psi(X)=[\text{deg}(\Psi(X))]$, I mean, how can I get the correspondence that corresponds to the dual isogeny of $X$,

b*) Is the last thing related with the self intersection number of $X$ ?