Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(1791\)\(\medspace = 3^{2} \cdot 199 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.1791.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | odd |
| Determinant: | 1.199.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.1791.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} - x^{2} - 2x + 1 \)
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The roots of $f$ are computed in $\Q_{ 313 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 166 + 201\cdot 313 + 6\cdot 313^{2} + 25\cdot 313^{3} + 242\cdot 313^{4} +O(313^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 196 + 298\cdot 313 + 173\cdot 313^{2} + 281\cdot 313^{3} + 231\cdot 313^{4} +O(313^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 269 + 292\cdot 313 + 37\cdot 313^{2} + 307\cdot 313^{3} + 247\cdot 313^{4} +O(313^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 309 + 145\cdot 313 + 94\cdot 313^{2} + 12\cdot 313^{3} + 217\cdot 313^{4} +O(313^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $3$ | |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ | |
| $6$ | $2$ | $(1,2)$ | $1$ | ✓ |
| $8$ | $3$ | $(1,2,3)$ | $0$ | |
| $6$ | $4$ | $(1,2,3,4)$ | $-1$ |