Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(10800\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.2.10800.2 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4$ |
Parity: | odd |
Determinant: | 1.3.2t1.a.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.2.10800.2 |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - 2x^{3} - 8x + 4 \) . |
The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 69 + 58\cdot 103 + 72\cdot 103^{2} + 85\cdot 103^{3} +O(103^{5})\) |
$r_{ 2 }$ | $=$ | \( 75 + 39\cdot 103 + 29\cdot 103^{2} + 15\cdot 103^{3} + 68\cdot 103^{4} +O(103^{5})\) |
$r_{ 3 }$ | $=$ | \( 77 + 20\cdot 103 + 59\cdot 103^{2} + 36\cdot 103^{3} + 53\cdot 103^{4} +O(103^{5})\) |
$r_{ 4 }$ | $=$ | \( 90 + 86\cdot 103 + 44\cdot 103^{2} + 68\cdot 103^{3} + 83\cdot 103^{4} +O(103^{5})\) |
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.