To try to put into words and what I was thinking.
I was wonder what is the difference between an elliptic curve defined over $F_p : E(F_p)$
And the quotient field which is defined by F_p over an elliptic curve equation. Is this a quotient field : $F_p[X] / (Y^2 = X^3 + AX + B)$?
What I believe I know is this:
$E(F_p)$ : The number of points in this set is found by using schoofs algorithm.
$F_p[X] / (Y^2 = X^3 + AX + B)$ : The number of points in this set is $p^3$ , 3 because an elliptic curves highest degree is 3.
Therefore, they are not the same.
Maybe there are more things which I am missing?
I am thinking that $F_p[X] / (Y^2 = X^3 + AX + B)$ is the same as just saying $F_p[X] / X^3$