I hope this question fits here.

Let $H/k$ be a genus $2$ curve and $J$ its Jacobian variety. Since $J(k)\cong \text{Pic}^0(H)(k)$ we have that its generic point looks like $[(x_1,y_1)+(x_2,y_2)-2\infty]\in J$. In Mumford coordinates we can see it as $g:=\langle x^2 -Ax + B,Cx+D\rangle:=\langle u(x),v(x)\rangle\in J$ with $u(x_i)=0$ and $v(x_i)=y_i$. This means that $A:=x_1+x_2, B:=x_1x_2, C:=\tfrac{y_1-y_2}{x_1-x_2}, D:=\tfrac{x_2y_1-x_1y_2}{x_1-x_2}$.

I calculated explicitly an element $\gamma\in\text{End}_k(J)$ using Mumford coordinates for the generic point, that is $\gamma(g)=<x^2 + \tfrac{A_1(g)}{A_2(g)}x + \tfrac{B_1(g)}{B_2(g)}, \tfrac{C_1(g)}{C_2(g)}x + \tfrac{ D_1(g)}{D_2(g)}>$ where $A_1,A_2,B_1,B_2,C_1,C_2,D_1,D_2\in k[J]$.

To formulate my question, note that since the endomorphism $\gamma$ works fine for the generic point, and $J$ has dimension $2$, we have that if for some $D\in J(k)$, its image under $\gamma$, namely $\gamma(D)$ is of the form $[(x_1,y_1)-\infty]$, then $\gamma$ won't be defined for $D$, since some of the denominators will be zero.

How can I distinguish when $\gamma(D)$ is 0 or when it is non-zero non-generic of the form $[(\tilde x,\tilde y)-\infty]$?

I know I could use a constant divisor $D_0$ and calculate $\gamma(D+D_0)$ when $\gamma(D)$ is not defined and then subtract with $\gamma(D_0)$ using Cantor addition. However, in my situation this is not possible since I am testing if $k$ is a field using some arithmetic geometry. I want to be able to use the functions defining $\gamma$ over $\mathbb{Q}$ to distinguish the situation of $\gamma(D)$ being $[0]$ or being of the form $[(\tilde x,\tilde y)-\infty]$

I have noted that when the ALL the denominators are 0, it looks like the image 0 in fact, but when it lies in the "theta divisor", (the image is a point of the form $[(x_1,y_1)-\infty]$), some of the denominators are non-zero. However I do not know how to distinguish this formally or maybe my examples are just "lucky" examples.

Is there a way with this information to distinguish when $\gamma(D)$ is exactly [0] ?

What I did

I tried to calculate the formula of $\gamma$ using MAGMA via the function field of $J$, using the usual relations for the Jacobian of $H$ plus the denominators of $\gamma$ as relations, but the computation does not finish and eats all my memory eventually.

I just need to know if a point in the image is 0 or non-generic when $\gamma(D)$ has 0's in the denominators using the information that I have, or maybe using an element of the function field of $J$ that can distinguish if a divisor is 0 or if it has the point at infinity with multiplicity 1.