Basic invariants
| Dimension: | $2$ |
| Group: | $S_3$ |
| Conductor: | \(1083\)\(\medspace = 3 \cdot 19^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 3.1.1083.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_3$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.1083.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 81\cdot 97 + 33\cdot 97^{2} + 85\cdot 97^{3} + 76\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 + 89\cdot 97 + 3\cdot 97^{2} + 86\cdot 97^{3} + 77\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 81 + 23\cdot 97 + 59\cdot 97^{2} + 22\cdot 97^{3} + 39\cdot 97^{4} +O(97^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,2)$ | $0$ |
| $2$ | $3$ | $(1,2,3)$ | $-1$ |