My question is as follows: Pick two values of a in $F_{11} = Z/11Z$ (a not equal to 3), such that the equation $y^2 = x^3+ax+1$ defines an elliptic curve (i.e., it is smooth).
For each such a, determine the number of points #E(F_11).
I'm new to elliptic curves in number theory, so any tips or solutions to this problem would be greatly appreciated!
For $a = 0$, we have the elliptic curve
$$y^2 \equiv x^3 + 1 \pmod{11} $$
A naive way to find the number of points is to just enumerate them, we find
$$ \begin{array}{c|lcr} x & \text{y's} \\ \hline 0 & 1, 10 \\ 1 & \text{None} \\ 2 & 3, 8 \\ 3 & \text{None} \\ 4 & \text{None} \\ 5 & 4, 7 \\ 6 & \text{None} \\ 7 & 5, 6 \\ 8 & \text{None} \\ 9 & 2, 9 \\ 10 & 0 \\ \end{array} $$
We can see that we have $11$ points in that table plus the point at infinity for a total of
$$\#E(F_{11}) = 12$$
You can find a second example.