Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(688\)\(\medspace = 2^{4} \cdot 43 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.688.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | odd |
| Determinant: | 1.43.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.688.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 2x - 1 \)
|
The roots of $f$ are computed in $\Q_{ 173 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 91 + 97\cdot 173 + 151\cdot 173^{2} + 23\cdot 173^{3} + 65\cdot 173^{4} +O(173^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 116 + 133\cdot 173 + 132\cdot 173^{2} + 27\cdot 173^{3} + 169\cdot 173^{4} +O(173^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 155 + 37\cdot 173 + 58\cdot 173^{2} + 151\cdot 173^{3} + 85\cdot 173^{4} +O(173^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 157 + 76\cdot 173 + 3\cdot 173^{2} + 143\cdot 173^{3} + 25\cdot 173^{4} +O(173^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ | |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ | |
| $6$ | $2$ | $(1,2)$ | $1$ | ✓ |
| $8$ | $3$ | $(1,2,3)$ | $0$ | |
| $6$ | $4$ | $(1,2,3,4)$ | $-1$ |