Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(127168\)\(\medspace = 2^{6} \cdot 1987 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.127168.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | even |
| Determinant: | 1.7948.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.2.127168.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - x^{4} + 4x^{3} - 2x^{2} + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 a + 28 + \left(25 a + 4\right)\cdot 29 + \left(18 a + 20\right)\cdot 29^{2} + \left(23 a + 19\right)\cdot 29^{3} + 24\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 a + 6 + \left(3 a + 3\right)\cdot 29 + \left(10 a + 2\right)\cdot 29^{2} + \left(5 a + 3\right)\cdot 29^{3} + \left(28 a + 5\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 a + 26 + \left(4 a + 9\right)\cdot 29 + \left(8 a + 19\right)\cdot 29^{2} + \left(12 a + 19\right)\cdot 29^{3} + 4\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 19 a + 18 + \left(24 a + 21\right)\cdot 29 + \left(20 a + 26\right)\cdot 29^{2} + \left(16 a + 14\right)\cdot 29^{3} + \left(28 a + 23\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 18 a + 4 + \left(16 a + 20\right)\cdot 29 + \left(21 a + 21\right)\cdot 29^{2} + \left(18 a + 7\right)\cdot 29^{3} + \left(8 a + 2\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 11 a + 7 + \left(12 a + 27\right)\cdot 29 + \left(7 a + 25\right)\cdot 29^{2} + \left(10 a + 21\right)\cdot 29^{3} + \left(20 a + 26\right)\cdot 29^{4} +O(29^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.