Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(1112\)\(\medspace = 2^{3} \cdot 139 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.154568.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.1112.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.8896.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 4x^{4} - 3x^{3} + 3x^{2} - 2x + 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 31 a + 19 + \left(33 a + 16\right)\cdot 41 + \left(11 a + 38\right)\cdot 41^{2} + 10 a\cdot 41^{3} + \left(13 a + 37\right)\cdot 41^{4} + \left(39 a + 19\right)\cdot 41^{5} + 8\cdot 41^{6} + \left(11 a + 25\right)\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 36 a + 20 + \left(29 a + 14\right)\cdot 41 + \left(9 a + 29\right)\cdot 41^{2} + \left(30 a + 13\right)\cdot 41^{3} + \left(15 a + 33\right)\cdot 41^{4} + \left(19 a + 4\right)\cdot 41^{5} + \left(3 a + 23\right)\cdot 41^{6} + \left(29 a + 5\right)\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 a + 30 + \left(7 a + 4\right)\cdot 41 + \left(29 a + 40\right)\cdot 41^{2} + \left(30 a + 19\right)\cdot 41^{3} + \left(27 a + 25\right)\cdot 41^{4} + \left(a + 1\right)\cdot 41^{5} + \left(40 a + 13\right)\cdot 41^{6} + \left(29 a + 16\right)\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 + 18\cdot 41 + 41^{2} + 11\cdot 41^{3} + 6\cdot 41^{4} + 35\cdot 41^{5} + 20\cdot 41^{6} + 34\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 a + 5 + \left(11 a + 27\right)\cdot 41 + \left(31 a + 28\right)\cdot 41^{2} + \left(10 a + 12\right)\cdot 41^{3} + \left(25 a + 9\right)\cdot 41^{4} + \left(21 a + 6\right)\cdot 41^{5} + \left(37 a + 14\right)\cdot 41^{6} + \left(11 a + 7\right)\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 + 41 + 26\cdot 41^{2} + 23\cdot 41^{3} + 11\cdot 41^{4} + 14\cdot 41^{5} + 2\cdot 41^{6} + 34\cdot 41^{7} +O(41^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-3$ |
| $3$ | $2$ | $(1,3)(4,6)$ | $-1$ |
| $3$ | $2$ | $(1,3)$ | $1$ |
| $6$ | $2$ | $(1,4)(3,6)$ | $1$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $-1$ |
| $8$ | $3$ | $(1,2,4)(3,5,6)$ | $0$ |
| $6$ | $4$ | $(1,6,3,4)$ | $1$ |
| $6$ | $4$ | $(1,6,3,4)(2,5)$ | $-1$ |
| $8$ | $6$ | $(1,6,5,3,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.