Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(5569611270384979\)\(\medspace = 17^{3} \cdot 10427^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.177259.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | odd |
| Determinant: | 1.177259.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.0.177259.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + x^{4} + 2x^{3} - x^{2} - x + 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{2} + 42x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a + 6 + \left(5 a + 24\right)\cdot 43 + \left(28 a + 10\right)\cdot 43^{2} + \left(17 a + 29\right)\cdot 43^{3} + \left(25 a + 20\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 41 a + 24 + \left(32 a + 34\right)\cdot 43 + \left(10 a + 17\right)\cdot 43^{2} + \left(42 a + 7\right)\cdot 43^{3} + \left(29 a + 7\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 22 + \left(10 a + 26\right)\cdot 43 + \left(32 a + 38\right)\cdot 43^{2} + 38\cdot 43^{3} + \left(13 a + 37\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 21 a + 23 + \left(24 a + 8\right)\cdot 43 + \left(3 a + 3\right)\cdot 43^{2} + \left(10 a + 14\right)\cdot 43^{3} + \left(5 a + 41\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 37 a + 12 + \left(37 a + 23\right)\cdot 43 + \left(14 a + 33\right)\cdot 43^{2} + \left(25 a + 18\right)\cdot 43^{3} + \left(17 a + 28\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 22 a + 1 + \left(18 a + 12\right)\cdot 43 + \left(39 a + 25\right)\cdot 43^{2} + \left(32 a + 20\right)\cdot 43^{3} + \left(37 a + 36\right)\cdot 43^{4} +O(43^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $15$ | $2$ | $(1,2)$ | $-1$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.