Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(229\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.229.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | even |
| Determinant: | 1.229.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.0.229.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x + 1 \)
|
The roots of $f$ are computed in $\Q_{ 193 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 135 + 139\cdot 193 + 151\cdot 193^{2} + 188\cdot 193^{3} + 63\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 145 + 66\cdot 193 + 184\cdot 193^{2} + 103\cdot 193^{3} + 30\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 148 + 174\cdot 193 + 160\cdot 193^{2} + 29\cdot 193^{3} + 160\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 151 + 4\cdot 193 + 82\cdot 193^{2} + 63\cdot 193^{3} + 131\cdot 193^{4} +O(193^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $3$ | |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ | ✓ |
| $6$ | $2$ | $(1,2)$ | $1$ | |
| $8$ | $3$ | $(1,2,3)$ | $0$ | |
| $6$ | $4$ | $(1,2,3,4)$ | $-1$ |