Twists arising from torsion points

Let $p$ be a prime number, $K$ a number field that contains the $p$-th root of unity ${\zeta }_p$, $d$ a $p$-power-free integer and $L=K(\sqrt[p]{d})$. Let $E\mathrm{/}K$ be an elliptic curve with full $p$-torsion and $S,T\in E(K)[p]$ be the generators. Define the cocycle ${\xi }_d:\mathrm{Gal}(\bar{K}\mathrm{/}K)\to E$ by $${\xi }_d(\sigma )=\{\begin{array}{ll}O,& \text{if }\sigma (\sqrt[p]{d})=\sqrt[p]{d},\\ kS,& \text{if }\sigma (\sqrt[p]{d})={\zeta }_p^k\sqrt[p]{d},\end{array}$$ and denote by $H_S^d$ the twist of $E$ corresponding to the cocycle ${\xi }_d$. In this paper we construct generators $z$ and $w$ of the function field $K(H_S^d)$ and give a model of the twist $$H_S^d:{\alpha }_1{z}^p+{\alpha }_2{z}^{p-2}w+\dots +{\alpha }_{\frac{p+1}{2}}z{w}^{\frac{p-1}{2}}+\beta w^p+\gamma =0.$$ We also obtain that the twist $H_S^d$ is everywhere locally solvable only for finitely many integers $d$.