Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(844\)\(\medspace = 2^{2} \cdot 211 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.178084.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.844.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.3376.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - x^{4} + x^{3} + x^{2} - 2x + 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 a + 13 + \left(3 a + 25\right)\cdot 29 + \left(15 a + 15\right)\cdot 29^{2} + \left(27 a + 11\right)\cdot 29^{3} + \left(17 a + 23\right)\cdot 29^{4} + \left(4 a + 2\right)\cdot 29^{5} + \left(22 a + 15\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 + 14\cdot 29 + 14\cdot 29^{2} + 11\cdot 29^{3} + 18\cdot 29^{4} + 18\cdot 29^{5} + 16\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 21 a + 12 + \left(28 a + 1\right)\cdot 29 + \left(25 a + 5\right)\cdot 29^{2} + \left(15 a + 17\right)\cdot 29^{3} + \left(15 a + 22\right)\cdot 29^{4} + \left(21 a + 12\right)\cdot 29^{5} + \left(4 a + 24\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 a + 28 + \left(25 a + 8\right)\cdot 29 + \left(13 a + 1\right)\cdot 29^{2} + \left(a + 18\right)\cdot 29^{3} + \left(11 a + 27\right)\cdot 29^{4} + \left(24 a + 7\right)\cdot 29^{5} + \left(6 a + 5\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a + 1 + 8\cdot 29 + \left(3 a + 19\right)\cdot 29^{2} + \left(13 a + 12\right)\cdot 29^{3} + \left(13 a + 26\right)\cdot 29^{4} + \left(7 a + 17\right)\cdot 29^{5} + \left(24 a + 26\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 + 28\cdot 29 + 29^{2} + 16\cdot 29^{3} + 26\cdot 29^{4} + 26\cdot 29^{5} + 27\cdot 29^{6} +O(29^{7})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-3$ |
| $3$ | $2$ | $(1,5)$ | $1$ |
| $3$ | $2$ | $(1,5)(2,6)$ | $-1$ |
| $6$ | $2$ | $(2,3)(4,6)$ | $1$ |
| $6$ | $2$ | $(1,5)(2,3)(4,6)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(1,6,5,2)$ | $1$ |
| $6$ | $4$ | $(1,4,5,3)(2,6)$ | $-1$ |
| $8$ | $6$ | $(1,6,4,5,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.