Basic invariants
| Dimension: | $2$ |
| Group: | $S_3$ |
| Conductor: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 3.1.108.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_3$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.108.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 9\cdot 31 + 4\cdot 31^{2} + 8\cdot 31^{3} + 23\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 + 13\cdot 31 + 29\cdot 31^{2} + 11\cdot 31^{3} + 25\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 + 8\cdot 31 + 28\cdot 31^{2} + 10\cdot 31^{3} + 13\cdot 31^{4} +O(31^{5})\)
|
Generators of the action on the roots $ r_{ 1 }, r_{ 2 }, r_{ 3 } $
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $ r_{ 1 }, r_{ 2 }, r_{ 3 } $ | Character values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,2)$ | $0$ |
| $2$ | $3$ | $(1,2,3)$ | $-1$ |