Take the 2-minute tour ×  I'm interested in Sklyanin Algebras or
Artin-Shelter regular algebras of type A. These are generated in
degree 1 by three variables x,y,z, and have three defining
relations in degree 2, which you can get by multiplying the
matrix $$M=\begin{pmatrix} cx & bz & ay \\ az & cy & bx \\ by &
ax & cz\end{pmatrix}$$ either on the left or right by the
row/column vector of variables. (For every triple of numbers
$a$, $b$, $c$, excluding a finite set, you get one of these
algebras.)  Such algebras $A$ contain a central regular element
of degree $3$, let me call it $g$. It has the property that
$A/gA$ is the twisted homogeneous coordinate ring of the scheme
of point modules of $A$, which is the cubic curve defined by
$\det M$.
$$g=c(c^3-b^3)y^3+b(c^3-a^3)yxz+a(b^3-c^3)xyz+c(a^3-c^3)x^3$$
This central element $g$ seems somewhat mysterious. In the
original paper by Artin-Shelter it was apparently found by a
computer search, in the foundational paper by Artin-Tate-Van den
Bergh, only its existence