A note on the procedure to find the generic polynomial of a quotient (closely following Adelmann)

There are 3 examples in these notes. The first one is the standard example of the cubic resolvent of a quartic. The second example is exactly from Adelmann \cite{Adelmann} and gives a defining polynomial corresponding to the unique $S_4$-quotient of $\mathrm{GL}_2(\mathbb{Z}/4\mathbb{Z})$. The splitting field of the Adelmann polynomial over $\mathbb{Q}$ is a subfield of the 4-division field of an elliptic curve, that contains the 2-division field of the elliptic curve. The third example is new and needed in the study of the field theory of quaternion origami. Associated to an elliptic curve defined over $\mathbb{Q}$, with a rational point, is a degree 8 polynomial whose Galois group is a subgroup of $\mathrm{Hol}(Q_8)$. Three defining polynomials corresponding to the three $S_4$-quotients of $\mathrm{Hol}(Q_8)$ are given.