Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(976\)\(\medspace = 2^{4} \cdot 61 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.2.976.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4$ |
Parity: | odd |
Determinant: | 1.244.2t1.a.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.2.976.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - 2x^{3} + 3x^{2} - 1 \) . |
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 10 + 14\cdot 43 + 25\cdot 43^{2} + 12\cdot 43^{3} + 25\cdot 43^{4} +O(43^{5})\) |
$r_{ 2 }$ | $=$ | \( 11 + 5\cdot 43 + 21\cdot 43^{2} + 21\cdot 43^{3} + 28\cdot 43^{4} +O(43^{5})\) |
$r_{ 3 }$ | $=$ | \( 26 + 35\cdot 43 + 20\cdot 43^{2} + 31\cdot 43^{3} + 37\cdot 43^{4} +O(43^{5})\) |
$r_{ 4 }$ | $=$ | \( 41 + 30\cdot 43 + 18\cdot 43^{2} + 20\cdot 43^{3} + 37\cdot 43^{4} +O(43^{5})\) |
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 | Complex conjugation |
$1$ | $1$ | $()$ | $3$ | |
$3$ | $2$ | $(1,2)(3,4)$ | $-1$ | |
$6$ | $2$ | $(1,2)$ | $1$ | ✓ |
$8$ | $3$ | $(1,2,3)$ | $0$ | |
$6$ | $4$ | $(1,2,3,4)$ | $-1$ |