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 symmetry of the maximum order and the next maximum order. Precisely, for the genus g hyperbolic surface with order 4g + 2 cyclic symmetry, the systole is $$\left( {1 + \cos {{\rm{\pi }} \over {2g + 1}} + \cos {{2{\rm{\pi }}} \over {2g + 1}}} \right)$$ when g ⩾ 7, and for the genus g hyperbolic surface with order 4g cyclic symmetry, the systole is $$\left( {1 + 2\cos {{\rm{\pi }} \over {2g}}} \right)$$ when g ⩾ 4.