Basic invariants
| Dimension: | $1$ |
| Group: | $C_2$ |
| Conductor: | $971 $ |
| Artin number field: | Splitting field of $f= x^{2} - x + 243 $ over $\Q$ |
| Size of Galois orbit: | 1 |
| Smallest containing permutation representation: | $C_2$ |
| Parity: | Odd |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 3 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ | $ 0 +O\left(3^{ 5 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 1 +O\left(3^{ 5 }\right)$ |
Generators of the action on the roots $ r_{ 1 }, r_{ 2 } $
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $ r_{ 1 }, r_{ 2 } $ | Character values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)$ | $-1$ |