Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(1936\)\(\medspace = 2^{4} \cdot 11^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.2.21296.1 |
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 4.2.21296.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - 2x^{3} - 4x^{2} - 6x - 2 \) . |
The roots of $f$ are computed in $\Q_{ 311 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 35 + 307\cdot 311 + 31\cdot 311^{2} + 273\cdot 311^{3} + 297\cdot 311^{4} +O(311^{5})\) |
$r_{ 2 }$ | $=$ | \( 52 + 304\cdot 311 + 34\cdot 311^{2} + 134\cdot 311^{3} + 78\cdot 311^{4} +O(311^{5})\) |
$r_{ 3 }$ | $=$ | \( 73 + 272\cdot 311 + 26\cdot 311^{2} + 301\cdot 311^{3} + 258\cdot 311^{4} +O(311^{5})\) |
$r_{ 4 }$ | $=$ | \( 153 + 49\cdot 311 + 217\cdot 311^{2} + 224\cdot 311^{3} + 297\cdot 311^{4} +O(311^{5})\) |
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.