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

Let a hyperelliptic curve $\mathcal{C}$ of genus $g$ defined over an algebraically closed field $K$ of characteristic $0$ be given by the equation $y^2=f(x)$, where $f(x)\in K[x]$ is a square-free polynomial of odd degree $2g+1$. The curve $\mathcal{C}$ contains a single “infinite” point $\mathcal{O}$, which is a Weierstrass point. There is a classical embedding of $\mathcal{C}(K)$ into the group $J(K)$ of $K$-points of the Jacobian variety $J$ of $\mathcal{C}$ that identifies the point $\mathcal{O}$ with the identity of the group $J(K)$. For $2\le g\le5$, we explicitly find representatives of birational equivalence classes of hyperelliptic curves $\mathcal{C}$ with a unique base point at infinity $\mathcal{O}$ such that the set $\mathcal{C}(K)\cap J(K)$ contains at least six torsion points of order $2g+1$. It was previously known that for $g=2$ there are exactly five such equivalence classes, and, for $g\ge3$, an upper bound depending only on the genus $g$ was known. We improve the previously known upper bound by almost $36$ times.