Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(1791\)\(\medspace = 3^{2} \cdot 199 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 4.2.1791.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4$ |
Parity: | odd |
Projective image: | $S_4$ |
Projective field: | Galois closure of 4.2.1791.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 313 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 166 + 201\cdot 313 + 6\cdot 313^{2} + 25\cdot 313^{3} + 242\cdot 313^{4} +O(313^{5})\)
$r_{ 2 }$ |
$=$ |
\( 196 + 298\cdot 313 + 173\cdot 313^{2} + 281\cdot 313^{3} + 231\cdot 313^{4} +O(313^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 269 + 292\cdot 313 + 37\cdot 313^{2} + 307\cdot 313^{3} + 247\cdot 313^{4} +O(313^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 309 + 145\cdot 313 + 94\cdot 313^{2} + 12\cdot 313^{3} + 217\cdot 313^{4} +O(313^{5})\)
| |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $3$ |
$3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
$6$ | $2$ | $(1,2)$ | $1$ |
$8$ | $3$ | $(1,2,3)$ | $0$ |
$6$ | $4$ | $(1,2,3,4)$ | $-1$ |