Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(1229\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.1229.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4$ |
Parity: | even |
Determinant: | 1.1229.2t1.a.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.0.1229.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} + 3x^{2} - x + 3 \) . |
The roots of $f$ are computed in $\Q_{ 151 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 29 + 133\cdot 151 + 5\cdot 151^{2} + 146\cdot 151^{3} + 124\cdot 151^{4} +O(151^{5})\) |
$r_{ 2 }$ | $=$ | \( 43 + 56\cdot 151 + 126\cdot 151^{2} + 17\cdot 151^{3} + 83\cdot 151^{4} +O(151^{5})\) |
$r_{ 3 }$ | $=$ | \( 105 + 63\cdot 151 + 39\cdot 151^{2} + 12\cdot 151^{3} + 65\cdot 151^{4} +O(151^{5})\) |
$r_{ 4 }$ | $=$ | \( 126 + 48\cdot 151 + 130\cdot 151^{2} + 125\cdot 151^{3} + 28\cdot 151^{4} +O(151^{5})\) |
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 |
$1$ | $1$ | $()$ | $3$ | |
$3$ | $2$ | $(1,2)(3,4)$ | $-1$ | ✓ |
$6$ | $2$ | $(1,2)$ | $1$ | |
$8$ | $3$ | $(1,2,3)$ | $0$ | |
$6$ | $4$ | $(1,2,3,4)$ | $-1$ |