Is there a single point generator G for every elliptic curve?

Question

I understand that G is a point on curve which holds true for y^2 = x^3 + ax + b. But there can be several other points as well which holds true for above equation but how do we choose which one is our G?

Answer 1

There often are multiple points which could be generators. In fact, for prime order curves, every point on the curve is a generator.

For practical use, a specific point is picked based on some constraints.

May curves use rules such as lowest abs(x) value or use some deterministic generator with a random seed.

I can't seem to find the login behind the G selection for secp256k1, however, but the point itself is defined in the curve specification.

Answer 2

It's defined in the specification of the curve. In Bitcoin, the curve was SECP256k1 is chosen and the curve parameters are defined here: SEC 2: Recommended Elliptic Curve Domain Parameters, Version 1.0