Let $A$ be a singular abelian surface over $\mathbb{C}$; that is, an abelian surface of maximal Picard rank $\rho(A)=4$. By Shioda-Mitani we know $A \cong E \times E'$ where $E,E'$ are isogenous elliptic curves with CM in an imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$. I'm not sure if this is standard terminology, but by the *effective semi-group*, I mean the semi-group $\text{NS}^{+}(A) \subset \text{NS}(A)$ of integral points in the effective cone of $A$.

We can take as a basis of $\text{NS}(A)$ the four classes $v, h, \Gamma, \Gamma_{\text{CM}}$, where $v,h$ are the vertical and horizontal classes in $E \times E'$, $\Gamma$ is the graph of an isogeny between $E, E'$, and $\Gamma_{\text{CM}}$ is the graph of the CM map. Obviously we get effective classes by taking non-negative integer linear combinations of these basis elements. However, $\text{NS}^{+}(A)$ is not finitely generated (see, page 1 of https://arxiv.org/pdf/alg-geom/9712019.pdf). So my questions are:

Do we have any understanding of the lattice points in $\text{NS}^{+}(A)$ which are not non-negative linear combinations of $v, h, \Gamma, \Gamma_{\text{CM}}$? Has this been studied anywhere? There are infinitely many such points, but I'm really lacking intuition for these.

Given an explicit class in $\text{NS}(A)$, is there any useful way of determining when it is effective? Other than the fact that it must intersect positively with an ample class. I haven't heard of such a condition in general, but I'm hoping maybe this particular case is easier.

Positivity in algebraic geomtery I, Lemma 1.5.4. This should at least provide some intuition, even if the CM case might be a bit harder. $\endgroup$