Basic invariants
Dimension: | $3$ |
Group: | $S_4$ |
Conductor: | \(3751\)\(\medspace = 11^{2} \cdot 31 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.2.3751.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4$ |
Parity: | odd |
Determinant: | 1.31.2t1.a.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.2.3751.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} - 2x^{2} - 3x - 2 \) . |
The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 7 + 68\cdot 149 + 93\cdot 149^{2} + 137\cdot 149^{3} + 149^{4} +O(149^{5})\)
$r_{ 2 }$ |
$=$ |
\( 71 + 14\cdot 149 + 7\cdot 149^{2} + 140\cdot 149^{3} + 47\cdot 149^{4} +O(149^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 76 + 45\cdot 149 + 79\cdot 149^{2} + 76\cdot 149^{3} + 66\cdot 149^{4} +O(149^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 145 + 20\cdot 149 + 118\cdot 149^{2} + 92\cdot 149^{3} + 32\cdot 149^{4} +O(149^{5})\)
| |
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.