the Klein-four group — which we denote by $G$ for the remainder of §\[subsec: examples\] — on $A$ by [$\mathbb{N}$]{}-graded algebra automorphisms. In Chapter \[chap: sklyanin\] we study a cocycle twist $A^{G,\mu}$ for a particular action of $G$ and a normalised 2-cocycle $\mu$. Some of the more surprising properties of $A^{G,\mu}$ are grouped together in parts (ii) and (iii) of the following result; it summarises Proposition \[claim: fatpoints\] and Theorems \[thm: 4sklytwistprops\] and \[prop: finitepointscheme\].

\[thm: sklyanin\] Let $E$ and $\sigma$ be the elliptic curve and automorphism associated to a 4-dimensional Sklyanin algebra $A$. There is an action of the Klein-four group $G$ on $A$ by [$\mathbb{N}$]{}-graded algebra automorphisms which, together with a normalised 2-cocycle $\mu$, produces a cocycle twist $A^{G,\mu}$ with the following properties:

-   it is strongly noetherian AS-regular domain of global dimension 4, which has Hilbert series $1/(1-t)^4$;

-   when $|\sigma| = \infty$ its point scheme consists of only 20 points (see Definition \[def: pointscheme\]);

-   it has a family of fat point modules of multiplicity 2 parameterised by $E^G$, an elliptic curve whose underlying structure comes from the orbit space of $E$ under a natural action of $G$.

The behaviour exhibited in Theorem \[thm: sklyanin\] (ii) 