Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(2151\)\(\medspace = 3^{2} \cdot 239 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.2.2151.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4$ |
Parity: | odd |
Determinant: | 1.239.2t1.a.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.2.2151.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{2} - 3x - 2 \) . |
The roots of $f$ are computed in $\Q_{ 113 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 4 + 101\cdot 113 + 64\cdot 113^{3} + 59\cdot 113^{4} +O(113^{5})\) |
$r_{ 2 }$ | $=$ | \( 61 + 50\cdot 113 + 40\cdot 113^{2} + 19\cdot 113^{3} + 28\cdot 113^{4} +O(113^{5})\) |
$r_{ 3 }$ | $=$ | \( 65 + 103\cdot 113 + 97\cdot 113^{2} + 46\cdot 113^{3} + 69\cdot 113^{4} +O(113^{5})\) |
$r_{ 4 }$ | $=$ | \( 96 + 83\cdot 113 + 86\cdot 113^{2} + 95\cdot 113^{3} + 68\cdot 113^{4} +O(113^{5})\) |
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.