Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(8860\)\(\medspace = 2^{2} \cdot 5 \cdot 443 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.15699920.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.8860.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.0.44300.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - 5x^{4} + 6x^{3} + 22x^{2} - 8x - 32 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 21 + 4\cdot 41^{2} + 40\cdot 41^{3} + 25\cdot 41^{4} + 23\cdot 41^{5} + 31\cdot 41^{6} + 41^{7} + 4\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 a + 1 + \left(18 a + 18\right)\cdot 41 + \left(31 a + 6\right)\cdot 41^{2} + \left(33 a + 14\right)\cdot 41^{3} + \left(22 a + 38\right)\cdot 41^{4} + \left(8 a + 17\right)\cdot 41^{5} + \left(25 a + 30\right)\cdot 41^{6} + \left(35 a + 36\right)\cdot 41^{7} + \left(38 a + 28\right)\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 26 + 22\cdot 41 + 39\cdot 41^{2} + 31\cdot 41^{3} + 38\cdot 41^{4} + 20\cdot 41^{5} + 37\cdot 41^{6} + 23\cdot 41^{7} + 28\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 28 a + 32 + \left(10 a + 10\right)\cdot 41 + \left(10 a + 26\right)\cdot 41^{2} + \left(27 a + 22\right)\cdot 41^{3} + \left(38 a + 31\right)\cdot 41^{4} + \left(10 a + 22\right)\cdot 41^{5} + \left(22 a + 37\right)\cdot 41^{6} + \left(38 a + 26\right)\cdot 41^{7} + \left(22 a + 1\right)\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 24 a + 11 + \left(22 a + 15\right)\cdot 41 + 9 a\cdot 41^{2} + \left(7 a + 2\right)\cdot 41^{3} + \left(18 a + 32\right)\cdot 41^{4} + \left(32 a + 20\right)\cdot 41^{5} + \left(15 a + 15\right)\cdot 41^{6} + \left(5 a + 36\right)\cdot 41^{7} + \left(2 a + 27\right)\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 13 a + 34 + \left(30 a + 14\right)\cdot 41 + \left(30 a + 5\right)\cdot 41^{2} + \left(13 a + 12\right)\cdot 41^{3} + \left(2 a + 38\right)\cdot 41^{4} + \left(30 a + 16\right)\cdot 41^{5} + \left(18 a + 11\right)\cdot 41^{6} + \left(2 a + 38\right)\cdot 41^{7} + \left(18 a + 31\right)\cdot 41^{8} +O(41^{9})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-3$ |
| $3$ | $2$ | $(1,3)$ | $1$ |
| $3$ | $2$ | $(1,3)(2,5)$ | $-1$ |
| $6$ | $2$ | $(2,4)(5,6)$ | $1$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $-1$ |
| $8$ | $3$ | $(1,2,4)(3,5,6)$ | $0$ |
| $6$ | $4$ | $(1,5,3,2)$ | $1$ |
| $6$ | $4$ | $(1,3)(2,6,5,4)$ | $-1$ |
| $8$ | $6$ | $(1,5,6,3,2,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.