Let $K$ be a field. Elliptic curve $E/K$ is defined as a genus 1 curve with base point $O$ (for example, in Silverman's book chapter 3).
But in this definition, what is a correct definition of base point ?
I have never seen mention about the definition of base point.
I think 'base point' refers to identity element as a group, or point at infinity of Weierstrass form, but these 'intuitive' explanation does not define correctly the notion 'base point'. It is not the same notion as rational point of $E$, how can we define 'base point' correctly ?