Basic invariants
| Dimension: | $9$ |
| Group: | $S_6$ |
| Conductor: | \(543498317107712\)\(\medspace = 2^{9} \cdot 101^{6} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.81608.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_{6}$ |
| Parity: | odd |
| Determinant: | 1.8.2t1.b.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.0.81608.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + x^{4} - x^{3} + 3x^{2} + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 109 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 109 }$:
\( x^{2} + 108x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 29 a + 41 + \left(10 a + 26\right)\cdot 109 + \left(96 a + 10\right)\cdot 109^{2} + \left(60 a + 6\right)\cdot 109^{3} + \left(81 a + 90\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 34 a + 69 + \left(25 a + 10\right)\cdot 109 + \left(98 a + 48\right)\cdot 109^{2} + \left(21 a + 83\right)\cdot 109^{3} + \left(20 a + 27\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 96 a + 29 + \left(92 a + 87\right)\cdot 109 + \left(63 a + 94\right)\cdot 109^{2} + \left(32 a + 90\right)\cdot 109^{3} + \left(84 a + 64\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 75 a + 103 + \left(83 a + 1\right)\cdot 109 + \left(10 a + 12\right)\cdot 109^{2} + \left(87 a + 7\right)\cdot 109^{3} + \left(88 a + 26\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 13 a + 16 + \left(16 a + 84\right)\cdot 109 + \left(45 a + 65\right)\cdot 109^{2} + \left(76 a + 59\right)\cdot 109^{3} + \left(24 a + 7\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 80 a + 70 + \left(98 a + 7\right)\cdot 109 + \left(12 a + 96\right)\cdot 109^{2} + \left(48 a + 79\right)\cdot 109^{3} + \left(27 a + 1\right)\cdot 109^{4} +O(109^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $9$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $40$ | $3$ | $(1,2,3)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.