Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(1264\)\(\medspace = 2^{4} \cdot 79 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.1264.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | even |
| Determinant: | 1.316.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.0.1264.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} + 3x^{2} - 2x + 1 \)
|
The roots of $f$ are computed in $\Q_{ 307 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 49 + 112\cdot 307 + 243\cdot 307^{2} + 73\cdot 307^{3} + 7\cdot 307^{4} +O(307^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 53 + 155\cdot 307 + 297\cdot 307^{2} + 221\cdot 307^{3} + 73\cdot 307^{4} +O(307^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 87 + 171\cdot 307 + 221\cdot 307^{2} + 125\cdot 307^{3} + 119\cdot 307^{4} +O(307^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 118 + 175\cdot 307 + 158\cdot 307^{2} + 192\cdot 307^{3} + 106\cdot 307^{4} +O(307^{5})\)
|
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.