Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(1255\)\(\medspace = 5 \cdot 251 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.2.1255.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4$ |
Parity: | odd |
Determinant: | 1.1255.2t1.a.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.2.1255.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{2} - 3x - 1 \) . |
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 16 + 13\cdot 43 + 35\cdot 43^{2} + 32\cdot 43^{3} + 36\cdot 43^{4} +O(43^{5})\) |
$r_{ 2 }$ | $=$ | \( 22 + 11\cdot 43 + 24\cdot 43^{2} + 21\cdot 43^{3} + 27\cdot 43^{4} +O(43^{5})\) |
$r_{ 3 }$ | $=$ | \( 23 + 40\cdot 43 + 32\cdot 43^{2} + 31\cdot 43^{3} + 36\cdot 43^{4} +O(43^{5})\) |
$r_{ 4 }$ | $=$ | \( 25 + 20\cdot 43 + 36\cdot 43^{2} + 42\cdot 43^{3} + 27\cdot 43^{4} +O(43^{5})\) |
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 | Complex conjugation |
$1$ | $1$ | $()$ | $3$ | |
$3$ | $2$ | $(1,2)(3,4)$ | $-1$ | |
$6$ | $2$ | $(1,2)$ | $1$ | ✓ |
$8$ | $3$ | $(1,2,3)$ | $0$ | |
$6$ | $4$ | $(1,2,3,4)$ | $-1$ |