Element structure of special linear group:SL(2,Q)

This article aims to discuss the element structure of special linear group:SL(2,Q), i.e., the special linear group of degree two over the field of rational numbers.

Related information

 * Element structure of special linear group of degree two over a field: This is very general, but requres a large number of cases and takes a lot of effort to understand in its entirety.
 * Element structure of special linear group of degree two over a finite field

Conjugacy class structure
The conjugacy class structure basically follows from Element structure of special linear group of degree two over a field and a number of computations specific to the rational numbers. Unfortunately, some of the actual descriptions of how conjugacy classes split relative to general linear group:GL(2,Q) requires deep (and may be open?) facts from number theory. However, we can fill in the broad outlines.