You got nothing wrong. To expand, you can add points on elliptic curves together. To add P and Q, you draw the line between them, find the third point on the li

You got nothing wrong. To expand, you can add points on elliptic curves together. To add P and Q, you draw the line between them, find the third point on the line, and reflect it over the x-axis [1]. Then the idea is to find a generating set S of points so that every rational point on the elliptic curve is a sum of points from S. The size of S is called the rank of the elliptic curve. [1] There are some details: https://en.wikipedia.org/wiki/Elliptic_curve#Group_law https://en.wikipedia.org/wiki/Elliptic_curve#Group_law