On hyperelliptic curves of odd degree and genus g with 6 torsion points of order 2g + 1

Let a hyperelliptic curve C of genus g defined over an algebraically closed field K of characteristic 0, given by the equation $y^2=f\left(x\right)$, where the polynomial $f\left(x\right)\in K\left[x\right]$ is square-free and has odd degree $2g+1$. The curve $C$ contains a single “infinite” point $O$, which is the Weierstrass point. There is a classical embedding of $C\left(K\right)$ into the group of $K$-points $J\left(K\right)$ of the Jacobian variety $J$ of the curve $C$, identifying the point $O$with the unit element of the group $J\left(K\right)$. For $2\le g\le 5$, the article explicitly found representatives of birational equivalence classes such hyperelliptic curves $C$ with a marked unique point at infinity $O$ that the set $C\left(K\right)\cap J\left(K\right)$ contains at least 6 torsion points of order $2g+1$. It was previously known that for $g=2$there are exactly 5 such equivalence classes, and for $g\ge 3$ an upper bound was known that depended only on the genus of $g$. We improve the previously known upper bound by almost 36 times.