Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(6809\)\(\medspace = 11 \cdot 619 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.4.6809.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | even |
| Determinant: | 1.6809.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.4.6809.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 5x^{2} - x + 1 \)
|
The roots of $f$ are computed in $\Q_{ 233 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 54 + 147\cdot 233 + 85\cdot 233^{2} + 81\cdot 233^{3} + 40\cdot 233^{4} +O(233^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 62 + 146\cdot 233 + 12\cdot 233^{2} + 214\cdot 233^{3} + 225\cdot 233^{4} +O(233^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 153 + 6\cdot 233 + 221\cdot 233^{2} + 221\cdot 233^{3} + 7\cdot 233^{4} +O(233^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 197 + 165\cdot 233 + 146\cdot 233^{2} + 181\cdot 233^{3} + 191\cdot 233^{4} +O(233^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 4 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 4 }$ | Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $6$ | $2$ | $(1,2)$ | $1$ |
| $8$ | $3$ | $(1,2,3)$ | $0$ |
| $6$ | $4$ | $(1,2,3,4)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.