I have a smooth, projective curve $X/mathbb{C}$ of genus $g$, embedded in a product of elliptic curves $A = prod_{i=1}^g E_i$. Since $H_*(A; mathbb{Z})$ with the Pontryagin product is isomorphic to the exterior algebra $Lambda(H_1(A; mathbb{Z})) cong Lambda(mathbb{Z}^{2g})$, given a basis $a_1, dots, a_{2g}$ for $H_1(A; mathbb{Z})$, we get a basis ${a_i wedge a_j : 1 leq i < j leq 2g}$ for $H_2(A; mathbb{Z})$. Is there a practical algorithm that computes the fundamental class $(X) in H_2(A; mathbb{Z})$ in terms of this basis, given defining equations for $X$ and each $E_i$? If so, has it been implemented in computer algebra software like Sage, Magma, Macaulay2, etc.?

(For broader context, my actual goal is to compute the degree of the induced map $operatorname{Jac}(X) to A$, which I’ve already proven is an isogeny in my case. If I can compute the fundamental class $(X) in H_2(A; mathbb{Z})$, then I can extract the degree of this isogeny from the value of the power $(X)^g$ under the Pontryagin product.)