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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. ...

This is hard, so I am looking for partial results and how hard it is. Let $n>4$. Is it true that the hyperelliptic curve $x^n=y(y+1)$ doesn't have rational point with $x \ne 0$? If necessarily ...

Probably this is known, but doesn't show in searches. If a certain hyperelliptic curve has only trivial rational points, FLT-like curve also has only trivial rationals points for fixed $n$. Working ...

Related to FLT and this question. For natural $n > 4 $ define the curve $C_n : z^{n-2}y(y+z)=x^n$. $C_n$ has the trivial points with $x=0$ for all $n$. The answer in the linked question shows ...