Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(1236544\)\(\medspace = 2^{6} \cdot 139^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 4.2.8896.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | even |
| Projective image: | $S_4$ |
| Projective field: | Galois closure of 4.2.8896.2 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 + 24\cdot 37 + 30\cdot 37^{2} + 19\cdot 37^{3} + 34\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 + 30\cdot 37 + 15\cdot 37^{2} + 12\cdot 37^{3} + 17\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 22 + 16\cdot 37 + 34\cdot 37^{2} + 26\cdot 37^{3} + 23\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 29 + 2\cdot 37 + 30\cdot 37^{2} + 14\cdot 37^{3} + 35\cdot 37^{4} +O(37^{5})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $6$ | $2$ | $(1,2)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)$ | $0$ |
| $6$ | $4$ | $(1,2,3,4)$ | $1$ |