 So if I want to be able to use elliptic curves for a cryptographic system, one of the things that will be very useful is if I have a list of the points that are on the elliptic curve. And so it helps if we work mod n. And so now let's consider my elliptic curves. And what we're going to do is we're going to change the problem slightly. If we're working mod n, then all of our points are going to have integer coordinates. So what I can look for then is the solutions that are in integers to the elliptic curve equation. And I could pick some modulus here, I'll pick 13. Now for reference, it'll be useful to know which numbers are perfect squares mod 13. So I'll go ahead and find that list, plus or minus 1 squared is 1, plus or minus 2 squared is 4, 9, those are easy, plus or minus 4 squared that's 16, mod 13 gives us 3, plus or minus 5 squared is 12, and so on. So unless x cubed plus xx minus 9 is one of these six numbers, then it's not going to be a perfect square, and I'm not going to have a point with integer coordinates mod 13. Well, let's take a look at that. At this point, it comes down to trial and error. If x equals 0, substituting that in y squared equals negative 9, which is the same as 4 mod 13, which is a perfect square. Solutions are y equals plus or minus 2, and so I have two integer points, x equals 0, y equals 2, x equals 0, y equals negative 2, except since I'm working mod 13, I don't have to and don't really want to work with negative numbers, so I'll change that to 0, 11. So that starts me on my list of integer points, 0, 2, 0, 11. I can try x equals 1, substituting that in, I get y squared equals congruent to 11, not a perfect square, so I'll try the next value, x equals 2, y squared congruent to 11, again not a perfect square, try the next value, x equals 3, gives me 10, which is a perfect square, mod 13, my solutions are plus or minus 6, and I have my two extra points. And I can continue in this fashion to find the remaining integer points, which gives me this list. And so now I have 4, 8, 12, 16 integer points.