Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(52441\)\(\medspace = 229^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 4.0.229.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4$ |
Parity: | even |
Projective image: | $S_4$ |
Projective field: | Galois closure of 4.0.229.1 |
Galois action
Roots of defining polynomial
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 values |
$c1$ | |||
$1$ | $1$ | $()$ | $3$ |
$3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
$6$ | $2$ | $(1,2)$ | $-1$ |
$8$ | $3$ | $(1,2,3)$ | $0$ |
$6$ | $4$ | $(1,2,3,4)$ | $1$ |