Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(6183\)\(\medspace = 3^{3} \cdot 229 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.4.1415907.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.687.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.0.229.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 5x^{4} - 2x^{3} + 7x^{2} + 5x - 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 25 a + 28 + \left(16 a + 1\right)\cdot 37 + \left(26 a + 1\right)\cdot 37^{2} + \left(30 a + 9\right)\cdot 37^{3} + \left(8 a + 7\right)\cdot 37^{4} + \left(34 a + 32\right)\cdot 37^{5} + \left(19 a + 14\right)\cdot 37^{6} + \left(25 a + 19\right)\cdot 37^{7} +O(37^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a + 17 + \left(20 a + 6\right)\cdot 37 + \left(10 a + 16\right)\cdot 37^{2} + \left(6 a + 31\right)\cdot 37^{3} + \left(28 a + 11\right)\cdot 37^{4} + \left(2 a + 12\right)\cdot 37^{5} + \left(17 a + 23\right)\cdot 37^{6} + \left(11 a + 27\right)\cdot 37^{7} +O(37^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 27 a + 8 + \left(35 a + 13\right)\cdot 37 + \left(28 a + 32\right)\cdot 37^{2} + \left(34 a + 34\right)\cdot 37^{3} + \left(26 a + 18\right)\cdot 37^{4} + \left(26 a + 29\right)\cdot 37^{5} + \left(23 a + 34\right)\cdot 37^{6} + \left(29 a + 2\right)\cdot 37^{7} +O(37^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 33 + 18\cdot 37 + 27\cdot 37^{2} + 12\cdot 37^{3} + 24\cdot 37^{4} + 9\cdot 37^{5} + 24\cdot 37^{6} + 4\cdot 37^{7} +O(37^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 20 + 15\cdot 37 + 32\cdot 37^{2} + 25\cdot 37^{3} + 30\cdot 37^{4} + 28\cdot 37^{5} + 21\cdot 37^{6} + 32\cdot 37^{7} +O(37^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 a + 5 + \left(a + 18\right)\cdot 37 + \left(8 a + 1\right)\cdot 37^{2} + \left(2 a + 34\right)\cdot 37^{3} + \left(10 a + 17\right)\cdot 37^{4} + \left(10 a + 35\right)\cdot 37^{5} + \left(13 a + 28\right)\cdot 37^{6} + \left(7 a + 23\right)\cdot 37^{7} +O(37^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-3$ |
| $3$ | $2$ | $(1,2)(3,6)$ | $-1$ |
| $3$ | $2$ | $(3,6)$ | $1$ |
| $6$ | $2$ | $(1,3)(2,6)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $8$ | $3$ | $(1,4,3)(2,5,6)$ | $0$ |
| $6$ | $4$ | $(1,6,2,3)$ | $1$ |
| $6$ | $4$ | $(1,2)(3,5,6,4)$ | $-1$ |
| $8$ | $6$ | $(1,4,3,2,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.