So I was reading this online article about twist curve and stuck at the following quote,
"...you are effectively working on a twist E‘ rather than E."
But what does it mean exactly? For supersingular curve $E$ and its twist $E'$, their abscissas form a partition in $\mathbb{F}_p$. Taking $(x,y)\in E(\mathbb F_{q^2})-E(\mathbb F_{q})$ and thus $(x,y')\in E'(\mathbb F_p)$, do we have something like $k(x,y)$ sharing the same $x$-coordinate with $k(x,y')$?