Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(316969\)\(\medspace = 563^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.563.1 |
| 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 4.2.563.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} + x^{2} - x - 1 \)
|
The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 80 + 67\cdot 137 + 75\cdot 137^{2} + 127\cdot 137^{3} + 122\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 108 + 29\cdot 137 + 103\cdot 137^{2} + 73\cdot 137^{3} + 31\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 111 + 67\cdot 137 + 30\cdot 137^{2} + 52\cdot 137^{3} + 26\cdot 137^{4} +O(137^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 113 + 108\cdot 137 + 64\cdot 137^{2} + 20\cdot 137^{3} + 93\cdot 137^{4} +O(137^{5})\)
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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.