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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. ...

Let $C$ be projective hyperelliptic curve over finite field $K$. What are bounds for the number of points $\#C(K)$? The Hasse-Weil bound requires smooth curves, and hyperelliptic curves are not smooth ...