Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.2.3888.1 |
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 4.2.3888.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - 2x^{3} - 6x + 3 \) . |
The roots of $f$ are computed in $\Q_{ 379 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 87 + 178\cdot 379 + 58\cdot 379^{2} + 322\cdot 379^{3} + 369\cdot 379^{4} +O(379^{5})\) |
$r_{ 2 }$ | $=$ | \( 149 + 144\cdot 379 + 5\cdot 379^{2} + 107\cdot 379^{3} + 158\cdot 379^{4} +O(379^{5})\) |
$r_{ 3 }$ | $=$ | \( 225 + 115\cdot 379 + 220\cdot 379^{2} + 184\cdot 379^{3} + 160\cdot 379^{4} +O(379^{5})\) |
$r_{ 4 }$ | $=$ | \( 299 + 319\cdot 379 + 94\cdot 379^{2} + 144\cdot 379^{3} + 69\cdot 379^{4} +O(379^{5})\) |
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.