On Lecacheux's family of quintic polynomials (Q2231987)

Let \({\mathbb{Q}}(p,r)\) be the rational function field over \({\mathbb{Q}}\) with two variables \(p,r\). Lecacheux's quintic polynomial \(Lec(p,r;X)\) is known as a \({\mathbb{Q}}\)-generic polynomial for the Frobenius group \(F_{20}\). The elliptic curve \({\mathcal{E}}_{p,r}\) associated to \(Lec(p,r;X)\) is defined. The curve \({\mathcal{E}}_{p,r}\) has an isogeny \(\nu\) od degree \(5\) over \({\mathbb{Q}}(p,r)\), and the \(5\)-division polynomial has a quadratic factor \(f_2(x)\) with a root \(\theta\). Then one obtains a point \(A\in{\mathcal{E}}_{p,r}(\overline{\mathbb{Q}}(p,r))\) of degree \(5\) with \(x(A)=\theta\), and \({\mathcal{E}}^*_{p,r}={\mathcal{E}}_{p,r}/<A>\). Let \(\nu^*:{\mathcal{E}}^*_{p,r}\to {\mathcal{E}}_{p,r}\) be the dual isogeny of \(\nu\). Let \(P=(x(P),y(P)\) be a \({\mathbb{Q}}\)-rational point on \({\mathcal{E}}_{p,r}\). Define the Lecacheux polynomial \(Lec(P;X)\) with respect to \(P\) by \(Lec(P;X)=Lec(p,\frac{x(P)}{4(p^2+4)W};X)\) where \(W\) is the quantity defined in terms of \(p\) and \(r\), and let \(Spl_{\mathbb{Q}}(Lec(P;X))\) be its splitting field over \({\mathbb{Q}}\). Theorem: For any \({\mathbb{Q}}\)-rational point \(P\in{\mathcal{E}}_{p,r}\), \(Lec(P;X)\) is irreducible over \({\mathbb{Q}}\) if and only if \(P\in\nu^*({\mathcal{E}}^*_{p,r}({\mathbb{Q}}))\); (ii) There is a bijection between the finite sets \{ subgroup of order \(5\) in \({\mathcal{E}}_{p,r}({\mathbb{Q}})/\nu^*({\mathcal{E}}_{p,r}({\mathbb{Q}}))\)\,\} and \{ \(Spl_{\mathbb{Q}}(Lec(P;X))\)\,|\, \(P\in{\mathcal{E}}_{p,r}({\mathbb{Q}})\setminus\nu^*({\mathcal{E}}^*_{p,r}({\mathbb{Q}}))\)\,\}. A number of examples are computed with SAGE illustrating the results. This generalizes the work of Brumer for \(D_5\) polynomials.