Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(90508\)\(\medspace = 2^{2} \cdot 11^{3} \cdot 17 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.362032.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.187.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.1538636.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 6x^{4} - x^{3} + 6x^{2} - x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 26 a + 7 + \left(36 a + 1\right)\cdot 41 + \left(32 a + 34\right)\cdot 41^{2} + \left(23 a + 28\right)\cdot 41^{3} + \left(36 a + 27\right)\cdot 41^{4} + \left(14 a + 19\right)\cdot 41^{5} + \left(2 a + 2\right)\cdot 41^{6} + 34\cdot 41^{7} + \left(30 a + 16\right)\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 + 17\cdot 41 + 32\cdot 41^{2} + 35\cdot 41^{3} + 39\cdot 41^{4} + 23\cdot 41^{5} + 38\cdot 41^{6} + 23\cdot 41^{7} + 6\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 a + 3 + \left(4 a + 3\right)\cdot 41 + \left(8 a + 14\right)\cdot 41^{2} + \left(17 a + 26\right)\cdot 41^{3} + \left(4 a + 31\right)\cdot 41^{4} + \left(26 a + 27\right)\cdot 41^{5} + \left(38 a + 35\right)\cdot 41^{6} + \left(40 a + 31\right)\cdot 41^{7} + \left(10 a + 24\right)\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a + 30 + \left(39 a + 30\right)\cdot 41 + \left(16 a + 22\right)\cdot 41^{2} + \left(36 a + 19\right)\cdot 41^{3} + \left(34 a + 17\right)\cdot 41^{4} + \left(16 a + 35\right)\cdot 41^{5} + \left(17 a + 23\right)\cdot 41^{6} + 7 a\cdot 41^{7} + \left(35 a + 15\right)\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 + 10\cdot 41 + 26\cdot 41^{2} + 23\cdot 41^{3} + 2\cdot 41^{4} + 6\cdot 41^{5} + 4\cdot 41^{6} + 27\cdot 41^{7} + 28\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 35 a + 7 + \left(a + 19\right)\cdot 41 + \left(24 a + 34\right)\cdot 41^{2} + \left(4 a + 29\right)\cdot 41^{3} + \left(6 a + 3\right)\cdot 41^{4} + \left(24 a + 10\right)\cdot 41^{5} + \left(23 a + 18\right)\cdot 41^{6} + \left(33 a + 5\right)\cdot 41^{7} + \left(5 a + 31\right)\cdot 41^{8} +O(41^{9})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $3$ | $2$ | $(1,6)(2,5)$ | $-1$ |
| $6$ | $2$ | $(2,3)(4,5)$ | $-1$ |
| $6$ | $2$ | $(1,6)(2,3)(4,5)$ | $1$ |
| $8$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,5,6,2)$ | $-1$ |
| $6$ | $4$ | $(1,6)(2,4,5,3)$ | $1$ |
| $8$ | $6$ | $(1,5,4,6,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.