This question is related to a previous question posted by me here answered by Prof. M. Stoll. 5-Descent or ($\sqrt{5}$-Descent?) on certain genus 2 Jacobians.

Here I ask some technicalities of a similar detailed question.

Let $\mathcal{J}$ be the Jacobian of the curve $y^2 = x^5 + 10$ and consider a prime of the form $\lambda_n=4\cdot 5^n - 1$. It is easy to see that $\text{End}(\mathcal{J})\cong \mathbb{Z}[\zeta_5]$. Moreover, $\zeta_5+\zeta_5^4=\tfrac{-1+\sqrt{5}}{2}$ and we can obtain explicitly the endomorphism $[\sqrt{5}]\in\text{End}_{\mathbb{F}_{\lambda_n}}(\mathcal{J})$ .

I am interested in checking for which $n$, the point $[(-1,3)-\infty]\in\mathcal{J}(\mathbb{F}_{\lambda_n})$ is in the image of $[\sqrt{5}]\in\text{End}_{\mathbb{F}_{\lambda_n}(\mathcal{J})}$.

In my first attempt I tried reading Michael Stoll's papers on the arithmetic of curves of the form $y^2 = x^\ell + A$ in order to try an explicit descent map of $[\sqrt{5}]$ using the Selmer group associated to $[\sqrt{5}]$. This approach in some moment starts to be very complicated.

Another try was using the explicit endomorphism $[\sqrt{5}]$ over $\mathbb{Q}(\zeta_5)$ obtained with the help of MAGMA. I tried to use the denominators of the expressions involving it to solve them to get $\infty$ and $(-1,3)$ modulo $\lambda_n$, that is, to obtain the divisor $[(-1,3)-\infty]$ (this approach was also recommended by Prof. Stoll for the endomorphism $\zeta_5+\zeta_5^4$, but for $\sqrt{5}$ is too intense for MAGMA). The problem is that MAGMA is unable to find the reduced components of these big expressions even with particular examples. I need them since I want to work with the scheme associated to this reduced components and add them to the Function field of the Jacobian to get equations of the generic point.

Do you have any other suggestion ?