Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(3481\)\(\medspace = 59^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.2.205379.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4$ |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.2.205379.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} - 7x^{2} + 11x + 3 \) . |
The roots of $f$ are computed in $\Q_{ 163 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 84 + 130\cdot 163 + 85\cdot 163^{2} + 26\cdot 163^{3} + 89\cdot 163^{4} +O(163^{5})\) |
$r_{ 2 }$ | $=$ | \( 105 + 41\cdot 163 + 149\cdot 163^{2} + 136\cdot 163^{3} + 18\cdot 163^{4} +O(163^{5})\) |
$r_{ 3 }$ | $=$ | \( 146 + 84\cdot 163 + 102\cdot 163^{2} + 69\cdot 163^{3} + 4\cdot 163^{4} +O(163^{5})\) |
$r_{ 4 }$ | $=$ | \( 155 + 68\cdot 163 + 151\cdot 163^{2} + 92\cdot 163^{3} + 50\cdot 163^{4} +O(163^{5})\) |
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.