Systoles of hyperbolic surfaces with big cyclic symmetry

We obtain the exact values of the systoles of these hyperbolic surfaces of genus $g$ with cyclic symmetries of the maximum order and the next maximum order. Precisely: for genus $g$ hyperbolic surface with order $4g+2$ cyclic symmetry, the systole is $2$\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}$ (1+\cos \frac{\pi}{2g+1}+\cos \frac{2\pi}{2g+1})$when$g\ge 7$,andforgenus$g$ hyperbolic surface with order $4g$ cyclic symmetry, the systole is $2$\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}$ (1+2\cos \frac{\pi}{2g})$when$g\ge4$.