Example of Constant Morphism

Question

I am reading Basic Theory of Elliptic Curves, there I came about a statement saying :

A Morphism of curves is either Surjective or Constant.

While studying Isogenes I came across examples of Surjective morphism but I am still wondering for an example of Constant morphism.

Answer 1

Let $E_1, E_2$ be elliptic curves. Fix a point $Q \in E_2$. Then the map $\phi : E_1 \rightarrow E_2$ defined by $P \mapsto Q$ is a constant morphism.

This is because it is a rational map (it is just effectively a constant polynomial) and is defined everywhere trivially.