Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(47444544\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 41^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.2.13776.1 |
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 4.2.13776.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} + x^{2} - 6x + 3 \) . |
The roots of $f$ are computed in $\Q_{ 283 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 96 + 40\cdot 283 + 148\cdot 283^{2} + 245\cdot 283^{3} + 95\cdot 283^{4} +O(283^{5})\) |
$r_{ 2 }$ | $=$ | \( 107 + 105\cdot 283 + 121\cdot 283^{2} + 88\cdot 283^{3} + 233\cdot 283^{4} +O(283^{5})\) |
$r_{ 3 }$ | $=$ | \( 164 + 230\cdot 283 + 262\cdot 283^{2} + 42\cdot 283^{3} + 216\cdot 283^{4} +O(283^{5})\) |
$r_{ 4 }$ | $=$ | \( 199 + 189\cdot 283 + 33\cdot 283^{2} + 189\cdot 283^{3} + 20\cdot 283^{4} +O(283^{5})\) |
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.