the group $\mathrm{C}_{5}$
- GAP SmallGroup id
- $[5, 1]$
- order
- $5$ (cyclic)
- structure
- $\mathrm{C}_{5}$
- presentation
- $\langle g_{1} \mid g_{1}^{5} \rangle$
character table
Columns are the conjugacy classes: order is the order of a class representative, size the number of elements in the class. Rows $\chi_i$ are the irreducible characters.
| order | 1 | 5 | 5 | 5 | 5 |
|---|---|---|---|---|---|
| size | 1 | 1 | 1 | 1 | 1 |
| $\chi_{ 1 }$ | $1$ | $1$ | $1$ | $1$ | $1$ |
| $\chi_{ 2 }$ | $1$ | $\zeta_{5}$ | $\zeta_{5}^{2}$ | $\zeta_{5}^{3}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
| $\chi_{ 3 }$ | $1$ | $\zeta_{5}^{2}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}$ | $\zeta_{5}^{3}$ |
| $\chi_{ 4 }$ | $1$ | $\zeta_{5}^{3}$ | $\zeta_{5}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{2}$ |
| $\chi_{ 5 }$ | $1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{3}$ | $\zeta_{5}^{2}$ | $\zeta_{5}$ |
hyperelliptic quotients
- dimension 3, representation $\operatorname{diag}(1, \zeta_{5}, \zeta_{5}^{2})$ with $q = 1$, $\operatorname{ord} \omega_X = 5$, moduli $1$
- dimension 4, representation $\operatorname{diag}(1, 1, \zeta_{5}, \zeta_{5}^{2})$ with $q = 2$, $\operatorname{ord} \omega_X = 5$, moduli $4$