Question

Is there another method to find points with rational coordinates on elliptic curves that involves using tangent lines?

Answer 1

There is another method called the tangent and chord method to find points with rational coordinates on elliptic curves.

Step-by-step explanation:

Yes, there is another method to find points with rational coordinates on elliptic curves that involves using tangent lines. This method is known as the tangent and chord method.

To find rational points on an elliptic curve using the tangent and chord method, follow these steps:

1. Start with a known rational point P on the curve.
2. Draw a line tangent to the curve at point P.
3. Find the intersection point Q of this tangent line and the curve.
4. Construct a line passing through the known point P and the newly found point Q.
5. This line will intersect the curve at another point R.
6. Repeat steps 2-5 with the new point R as the known point until you have found all rational points on the curve.

This method allows for the discovery of additional rational points on the curve by using the tangent line as a guide. It is a useful technique for exploring the properties and structure of elliptic curves.