  In this note, we classify the conjugacy classes of
$\widetilde{\mathrm{SL}}_2(\mathbb{R})$, the universal covering group of
$\mathrm{PSL}_2(\mathbb{R})$. For any non-central element $\alpha \in
\widetilde{\mathrm{SL}}_2(\mathbb{R})$, we show that its conjugacy class may be
determined by three invariants:
  (i) Trace: the trace (valued in the set of positive real numbers
$\mathbb{R}_{+}$) of its image $\overline{\alpha}$ in
$\mathrm{PSL}_2(\mathbb{R})$;
  (ii) Direction type: the sign behavior of the induced self-homeomorphism of
$\mathbb{R}$ determined by the lifting $\widetilde{\mathrm{SL}}_2(\mathbb{R})
\curvearrowright \mathbb{R}$ of the action $\mathrm{PSL}_2(\mathbb{R})
\curvearrowright \mathbb{S}^{1}$;
  (iii) The function $\ell^{\sharp}$: a conjugacy invariant length function
introduced by S. Mochizuki [Res. Math. Sci. 3 (2016), 3:6].
