Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(1593\)\(\medspace = 3^{3} \cdot 59 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 4.0.1593.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4$ |
Parity: | even |
Projective image: | $S_4$ |
Projective field: | Galois closure of 4.0.1593.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 5 + 11\cdot 149 + 90\cdot 149^{2} + 23\cdot 149^{3} + 41\cdot 149^{4} +O(149^{5})\) |
$r_{ 2 }$ | $=$ | \( 32 + 96\cdot 149 + 102\cdot 149^{2} + 23\cdot 149^{3} + 49\cdot 149^{4} +O(149^{5})\) |
$r_{ 3 }$ | $=$ | \( 52 + 115\cdot 149 + 21\cdot 149^{2} + 92\cdot 149^{3} + 14\cdot 149^{4} +O(149^{5})\) |
$r_{ 4 }$ | $=$ | \( 62 + 75\cdot 149 + 83\cdot 149^{2} + 9\cdot 149^{3} + 44\cdot 149^{4} +O(149^{5})\) |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $3$ |
$3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
$6$ | $2$ | $(1,2)$ | $1$ |
$8$ | $3$ | $(1,2,3)$ | $0$ |
$6$ | $4$ | $(1,2,3,4)$ | $-1$ |