I have the following hyperelliptic curve of genus $2$:
$$
y^2 = 561 x^6 - 41904 x^5 + 627264 x^4 + 11860992 x^3 - 197074944 x^2 + 124416^2
$$
I need to find all the rational points on this curve. There is one obvious point at $(0, 124416)$. I have some experience in finding rational points on elliptic curves. I also have the reference *Handbook of Elliptic and Hyperelliptic Curve Cryptography* (Discrete Mathematics and Its Applications). Most of this work uses genus $2$ curves reduced to quintic form. So a related question is how to transform this sextic to the quintic form.

Thank you

Lorenz Menke

A question as to where this problem originates. The nuclear electric quadrupole Hamiltonian characteristic equation for spin $9/2$ is $$ E_p^5 - 99 (3 + z)E_p^3 - 1188(1 - z)E_p^2 + 1188(3 + z)^2 E_p + 11664(3 + z)(1 - z) $$ where the energy states are $E = 2 E_p$ and $z = \eta^2$ where $\eta$ is the asymmetry parameter. Now solve this equation for $z$. The above sextic is the square root term. Thus when the sextic is a perfect square for a rational energy state the resultant asymmetry parameter squared is also a rational. This set of rational points of this sextic are the values that result in the quadrupole quintic Galois factoring into a linear times a quartic. The only other values of $z$ that reduce this quintic are $z = 0, 1, -3,$ and $9$. The goal is to find all the points $z$ such that the quintic factors over the rationals.

ratpoints(suggested byFelipe Voloch) quickly finds 8 more rational point pairs, with $x = -1/6$, $1/12$, $-1/18$, $-1/24$, $1/36$, $1/48$, $-1/72$, and $11/216$. Taking $x=1/24X$ and clearing square factors simplifies the equation to $Y^2 = 561X^6-1746X^5+1089X^4+858X^3-594X^2+81$, when the small points are at $X = -3, 1, 0, 2, 1/2, 3/2, -3/4, 1/4, 9/11$, andratpointssays there are no others with $X =m/n$ with $|m|,|n| < 10^4$. $\endgroup$ – Noam D. Elkies Dec 2 '14 at 4:03