Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(133056\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 7 \cdot 11 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.2.133056.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4$ |
Parity: | odd |
Determinant: | 1.231.2t1.a.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.2.133056.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - 2x^{3} + 6x^{2} + 4x - 2 \) . |
The roots of $f$ are computed in $\Q_{ 197 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 50 + 196\cdot 197 + 181\cdot 197^{2} + 22\cdot 197^{3} + 56\cdot 197^{4} +O(197^{5})\)
$r_{ 2 }$ |
$=$ |
\( 54 + 157\cdot 197 + 196\cdot 197^{2} + 136\cdot 197^{3} + 7\cdot 197^{4} +O(197^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 126 + 90\cdot 197 + 25\cdot 197^{2} + 142\cdot 197^{3} + 79\cdot 197^{4} +O(197^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 166 + 146\cdot 197 + 186\cdot 197^{2} + 91\cdot 197^{3} + 53\cdot 197^{4} +O(197^{5})\)
| |
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.