Let $E$ be an elliptic curve defined over $\mathbb{Q}$ by the equation $$E: y^2=x^3-Ax+B=:f(x).$$

Consider the abelian variety $E^3:=E \times E \times E \subset \mathbb{P}^2 \times \mathbb{P}^2 \times \mathbb{P}^2$. Let $(x_1,y_1,x_2,y_2,x_3,y_3)$ denote the affine coordinates for a point in $E^3$. Inside $E^3$ we consider the curve $\mathcal{C}$ defined by the equations

$$\mathcal{C}:\begin{cases} y_1^2=f(x_1);\\ y_2^2=f(x_2);\\ y_3^2=f(x_3);\\ y_1=x_2^m;\\ y_2=x_3^n, \end{cases}$$ where $m$ and $n$ are positive integers, which can be taken as large as desired, if necessary.

I would like to know how one can show that $\mathcal{C}$ is not contained in any abelian subvariety of $E^3$ of dimension two. I would hope that once $m$ and $n$ are sufficiently large, one can prove this, but I cannot make any progress with it at this stage.