  As an algebraic surface, the equation of $E_8$-singularity $x^5+y^3+z^2=0$
can be obtained from a quotient $C_Y/\text{SL}(2, 13)$ over the modular curve
$X(13)$, where $Y \subset \mathbb{CP}^5$ is a complete intersection curve given
by a system of $\text{SL}(2, 13)$-invariant polynomials and $C_Y$ is a cone
over $Y$. It is different from the Kleinian singularity $\mathbb{C}^2/\Gamma$,
where $\Gamma$ is the binary icosahedral group. This gives a negative answer to
Arnol'd and Brieskorn's questions about the mysterious relation between the
icosahedron and $E_8$, i.e., the $E_8$-singularity is not necessarily the
Kleinian icosahedral singularity. In particular, the equation of
$E_8$-singularity possesses infinitely many kinds of distinct modular
parametrizations, and there are infinitely many kinds of distinct constructions
of the $E_8$-singularity. They form a variation of the $E_8$-singularity
structure over the modular curve $X(13)$, for which we give its algebraic
version, geometric version, $j$-function version and the version of
Poincar\'{e} homology $3$-sphere as well as its higher dimensional lifting,
i.e., Milnor's exotic $7$-sphere. Moreover, there are variations of $Q_{18}$
and $E_{20}$-singularity structures over $X(13)$. Thus, three different
algebraic surfaces, the equations of $E_8$, $Q_{18}$ and $E_{20}$-singularities
can be realized from the same quotients $C_Y/\text{SL}(2, 13)$ over the modular
curve $X(13)$ and have the same modular parametrizations.
