For an elliptic curve $E$, I understand that the notation $End(E)$ denotes the ring of endomorphisms of $E$. Since End(E) is torsion free, it's possible to take $\alpha, \beta \in End(E)$ and construct objects of the form $\frac{\alpha}{\beta} \cong End(E)\otimes_{\mathbb{Z}} \mathbb{Q}$. Many authors denote this division ring by $End^0(E)$. (Example: Ben Smith's slide 9 related complex multiplication. I have seen it at numerous other places.)

My question is, what does the $0$ in $End^0(E)$ indicate? Also, are there other objects like $End^1(E)$, etc.?

$\newcommand{\End}{\operatorname{End}}$For an elliptic curve $E$, I understand that the notation $\End(E)$ denotes the ring of endomorphisms of $E$. Since $\End(E)$ is torsion free, it's possible to take $\alpha, \beta \in \End(E)$ and construct objects of the form $\frac{\alpha}{\beta} \cong \End(E)\otimes_{\mathbb{Z}} \mathbb{Q}$. Many authors denote this division ring by $\End^0(E)$. (Example: Ben Smith's slide 9 related complex multiplication. I have seen it at numerous other places.)

My question is, what does the $0$ in $\End^0(E)$ indicate? Also, are there other objects like $\End^1(E)$, etc.?