Given two elliptic curves over $\overline{\mathbb{F}}_p$ in the form of trivariate degree $3$ polynomial we want to find whether they have a common intersection point. This is a decision problem(I don't want to find the point of intersection.)
Its seems really easy when elliptic curves are given in their Weierstrass normal form that is : $$y^2=x^3+A_1x+B_1 \qquad y^2=x^3+A_2x+B_2$$ Clearly this will certainly have a common point when $A_1\neq A_2$.
My question is can we test this in general ??
Also can we say something about curves of higher genus ?
Note : I want an intersection point in $\overline{\mathbb{F}}_p$ and not in ${\mathbb{F}}_p$