This is regarding finding points along an Elliptic Curve
$y^2 = x^3 + 8x + 1$ Over $\Bbb F_{11}$
In the "Elliptic Curves over Finite Fields" chapter of "Elliptic Curves: Number Theory and Cryptography" by Lawrence C. Washington, all the points along an elliptic curve are found by using a table of all the possible $x$ values in the field (0 and all the positive integers up to $p-1$ where $p$ is the finite field order) and considering which of these values are square numbers.
Here is a picture of the example. To test my understanding, I have carried out a similar process with the Elliptic Curve and Galois Field given at the top.
I found all the points, or so I thought I did. Here are the values I found. I have only included them up to 4, but I have computed all the values elsewhere.
$$\begin{array}{|c|c|c|c|c|c|} \text{$x$} & \text{$x^3+8x+1$ $mod$ $11$} & \text{$y$} & \text{Coordinates $mod$ $11$} \\ \hline \text{0} & 1 & 1,-1 & (0,1), (0,10)\\ \hline \text{1} & 10 & - & - \\ \hline \text{2} & 3 & - & - \\ \hline \text{3} & 8 & - & - \\ \hline \text{4} & 9 & 3,-3 & (4,3), (4,8) \\ \hline \end{array}$$
I hence found all the points with this method. I then added the point $P=(0,1)$ to itself and found that 8P was equal to (2,5). However, in the table above, we saw that when $x=2$, $x^3+8x+1$ mod 11 was equal to 3, which is not a square number, so there would not be a point along the elliptic curve there. Of course, I noticed that $x^3+8x+1$ when $x=2$ was equal to 25, and 25 is a square number, giving $y$ values of 5 and -5.
I am confused as to why we square root 25 before we take the modulus. With the table I followed in Washington, the modulus was taken first and then the result was square rooted.
A similar case arises when $x=8$. Taking the modulus first gives 5, which is not a square number, so this was omitted from the table method. 6P gives a point with $x$-coordinate 8. At $x=8$, $x^3+8x+1=577$, and 577 mod 11 = 5 mod 11 = 16 mod 11. This means that $y$ is 4 or -4 because 16 is a square number.
577 is a less obvious example to spot than 25. How do we know when the number is a square number in the modulus, just by seeing the remainder? Is the best method to simply check if $x+11k$, where k is an integer, is a square number for each value of $x$?
To recap, my main question is, with this table method, how do we find the points along the elliptic curve which have y-coordinates which have a square greater than the order of the field? So, how do we know when $x$ mod p can be rewritten as $a^2$ mod p where a is an integer?