Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(8303\)\(\medspace = 19^{2} \cdot 23 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.2.8303.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4$ |
Parity: | odd |
Determinant: | 1.23.2t1.a.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.2.8303.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} - 3x^{2} + 4x - 3 \) . |
The roots of $f$ are computed in $\Q_{ 211 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 35 + 48\cdot 211 + 143\cdot 211^{2} + 120\cdot 211^{3} + 193\cdot 211^{4} +O(211^{5})\)
$r_{ 2 }$ |
$=$ |
\( 95 + 71\cdot 211 + 147\cdot 211^{2} + 111\cdot 211^{3} + 50\cdot 211^{4} +O(211^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 145 + 46\cdot 211 + 196\cdot 211^{2} + 7\cdot 211^{3} + 141\cdot 211^{4} +O(211^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 148 + 44\cdot 211 + 146\cdot 211^{2} + 181\cdot 211^{3} + 36\cdot 211^{4} +O(211^{5})\)
| |
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.