Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(559504\)\(\medspace = 2^{4} \cdot 11^{2} \cdot 17^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.2.50864.2 |
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 4.2.50864.2 |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - 2x^{3} - 8x^{2} - 8x - 1 \) . |
The roots of $f$ are computed in $\Q_{ 397 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 119 + 290\cdot 397 + 197\cdot 397^{2} + 366\cdot 397^{3} + 73\cdot 397^{4} +O(397^{5})\) |
$r_{ 2 }$ | $=$ | \( 160 + 173\cdot 397 + 5\cdot 397^{2} + 334\cdot 397^{3} + 381\cdot 397^{4} +O(397^{5})\) |
$r_{ 3 }$ | $=$ | \( 168 + 80\cdot 397 + 249\cdot 397^{2} + 327\cdot 397^{3} + 388\cdot 397^{4} +O(397^{5})\) |
$r_{ 4 }$ | $=$ | \( 349 + 249\cdot 397 + 341\cdot 397^{2} + 162\cdot 397^{3} + 346\cdot 397^{4} +O(397^{5})\) |
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.