Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(2096\)\(\medspace = 2^{4} \cdot 131 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.2096.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | odd |
| Determinant: | 1.131.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.2096.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 2x^{2} - 2x + 2 \)
|
The roots of $f$ are computed in $\Q_{ 353 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 226 + 79\cdot 353 + 63\cdot 353^{2} + 27\cdot 353^{3} + 341\cdot 353^{4} +O(353^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 227 + 115\cdot 353 + 144\cdot 353^{2} + 11\cdot 353^{3} + 197\cdot 353^{4} +O(353^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 265 + 241\cdot 353 + 353^{2} + 34\cdot 353^{3} + 91\cdot 353^{4} +O(353^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 341 + 268\cdot 353 + 143\cdot 353^{2} + 280\cdot 353^{3} + 76\cdot 353^{4} +O(353^{5})\)
|
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.