Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(76176\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 23^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 4.2.228528.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | even |
| Projective image: | $S_4$ |
| Projective field: | Galois closure of 4.2.228528.2 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 367 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 65 + 54\cdot 367 + 318\cdot 367^{2} + 348\cdot 367^{3} + 228\cdot 367^{4} +O(367^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 119 + 328\cdot 367 + 350\cdot 367^{2} + 32\cdot 367^{3} + 76\cdot 367^{4} +O(367^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 201 + 68\cdot 367 + 186\cdot 367^{2} + 65\cdot 367^{3} + 105\cdot 367^{4} +O(367^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 349 + 282\cdot 367 + 245\cdot 367^{2} + 286\cdot 367^{3} + 323\cdot 367^{4} +O(367^{5})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $6$ | $2$ | $(1,2)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)$ | $0$ |
| $6$ | $4$ | $(1,2,3,4)$ | $1$ |