Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(257\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.257.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4$ |
Parity: | even |
Determinant: | 1.257.2t1.a.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.0.257.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} + x^{2} - x + 1 \) . |
The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 73 + 47\cdot 157 + 11\cdot 157^{2} + 57\cdot 157^{3} + 146\cdot 157^{4} +O(157^{5})\) |
$r_{ 2 }$ | $=$ | \( 75 + 34\cdot 157 + 88\cdot 157^{2} + 46\cdot 157^{3} + 3\cdot 157^{4} +O(157^{5})\) |
$r_{ 3 }$ | $=$ | \( 77 + 138\cdot 157 + 50\cdot 157^{2} + 98\cdot 157^{3} + 122\cdot 157^{4} +O(157^{5})\) |
$r_{ 4 }$ | $=$ | \( 89 + 93\cdot 157 + 6\cdot 157^{2} + 112\cdot 157^{3} + 41\cdot 157^{4} +O(157^{5})\) |
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 |
$1$ | $1$ | $()$ | $3$ | |
$3$ | $2$ | $(1,2)(3,4)$ | $-1$ | ✓ |
$6$ | $2$ | $(1,2)$ | $1$ | |
$8$ | $3$ | $(1,2,3)$ | $0$ | |
$6$ | $4$ | $(1,2,3,4)$ | $-1$ |