I am currently revising for an exam and need some help with a question. Below is an example from my notes which I am trying to understand. I can fill out the table just fine, but I can't figure out how to find the points from it. If someone could run through getting each point I would be grateful.
Example 1: Let p = 5 and y2 = x3-3x^2+3x. Draw up a table of values of
x^3-3x^2+3x.
x x^2 x^3 -3x^2 3x x^3-3x^2+3x
0 0 0 0 0 0
1 1 1 2 3 1
2 4 3 3 1 2
3 4 2 3 4 4
4 1 4 2 2 3
Now mod p, 0^2=0, 1^2=1, 2^2=4, 3^2=4 and 4^2=1.
(please note that the equal signs in the above line are congruence signs in notes, I don't know how to put them in here)
Thus we can list the points of our elliptic curve mod p.
(0, 0) (1, 1) (1, 4) (3, 2) (3, 3)
You know that for say $x=1$, $y^2 = 1$. The solutions to this equation (mod 5) are $y = 4$ and $y=1$. The same goes for the other points. To repeat,in more generality, the only thing you need to do is to figure out what y has the property that $y^2= a^3-3a^2+3a$, where $a \in \mathbb{Z}/5\mathbb{Z}$. Say that $b \in \mathbb{Z}/5\mathbb{Z}$ has the property that $b^2 = a^3-3a^2+3a$. Then $(a,b)$ is a point on the curve.