Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(6851\)\(\medspace = 13 \cdot 17 \cdot 31 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.212381.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.6851.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.1514071.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - 4x^{4} + 7x^{3} + 4x^{2} - 2x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{2} + 42x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 37 a + 39 + \left(33 a + 37\right)\cdot 43 + \left(42 a + 5\right)\cdot 43^{2} + \left(4 a + 12\right)\cdot 43^{3} + \left(2 a + 28\right)\cdot 43^{4} + \left(14 a + 27\right)\cdot 43^{5} + \left(21 a + 18\right)\cdot 43^{6} + \left(38 a + 35\right)\cdot 43^{7} +O(43^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 32 a + 7 + \left(17 a + 35\right)\cdot 43 + \left(13 a + 17\right)\cdot 43^{2} + \left(12 a + 31\right)\cdot 43^{3} + \left(17 a + 8\right)\cdot 43^{4} + \left(28 a + 12\right)\cdot 43^{5} + \left(3 a + 2\right)\cdot 43^{6} + \left(2 a + 22\right)\cdot 43^{7} +O(43^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 a + 33 + \left(9 a + 34\right)\cdot 43 + 14\cdot 43^{2} + \left(38 a + 17\right)\cdot 43^{3} + \left(40 a + 25\right)\cdot 43^{4} + \left(28 a + 39\right)\cdot 43^{5} + \left(21 a + 25\right)\cdot 43^{6} + \left(4 a + 9\right)\cdot 43^{7} +O(43^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 a + 39 + \left(25 a + 20\right)\cdot 43 + \left(29 a + 13\right)\cdot 43^{2} + \left(30 a + 30\right)\cdot 43^{3} + \left(25 a + 13\right)\cdot 43^{4} + \left(14 a + 23\right)\cdot 43^{5} + \left(39 a + 20\right)\cdot 43^{6} + \left(40 a + 20\right)\cdot 43^{7} +O(43^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 6 + 42\cdot 43 + 11\cdot 43^{2} + 41\cdot 43^{3} + 18\cdot 43^{4} + 29\cdot 43^{5} + 12\cdot 43^{6} + 10\cdot 43^{7} +O(43^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 + 43 + 22\cdot 43^{2} + 39\cdot 43^{3} + 33\cdot 43^{4} + 39\cdot 43^{5} + 5\cdot 43^{6} + 31\cdot 43^{7} +O(43^{8})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $3$ | $2$ | $(2,3)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $6$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $8$ | $3$ | $(1,5,2)(3,4,6)$ | $0$ |
| $6$ | $4$ | $(1,3,4,2)$ | $1$ |
| $6$ | $4$ | $(1,4)(2,6,3,5)$ | $-1$ |
| $8$ | $6$ | $(1,5,2,4,6,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.