  We study origami $f: C \rightarrow E$ with $G$-Galois cover $Q_8$. For a
point $P \in E(\mathbb{Q}) \backslash \left\{ \mathcal{O} \right\}$, we study
the field obtained by adjoining to $\mathbb{Q}$ the coordinates of all of the
preimages of $P$ under $f$. We find a defining polynomial, $f_{E, Q_8,P}$, for
this field and study its Galois group. We give an isomorphism depending on $P$
between a certain subfield of this field and a certain subfield of the
4-division field of the elliptic curve.
