Basic invariants
| Dimension: | $9$ |
| Group: | $S_6$ |
| Conductor: | \(21883862619611\)\(\medspace = 83^{3} \cdot 337^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.27971.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_{6}$ |
| Parity: | odd |
| Determinant: | 1.27971.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.0.27971.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + x^{4} - 2x^{3} + 3x^{2} - 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 107 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 107 }$:
\( x^{2} + 103x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 44 a + 97 + \left(12 a + 63\right)\cdot 107 + \left(16 a + 63\right)\cdot 107^{2} + \left(56 a + 99\right)\cdot 107^{3} + \left(35 a + 49\right)\cdot 107^{4} +O(107^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 43 + 80\cdot 107 + 9\cdot 107^{2} + 19\cdot 107^{3} + 94\cdot 107^{4} +O(107^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 81 + 2\cdot 107^{2} + 34\cdot 107^{3} + 96\cdot 107^{4} +O(107^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 + 10\cdot 107 + 23\cdot 107^{2} + 44\cdot 107^{3} + 27\cdot 107^{4} +O(107^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 39 + 96\cdot 107 + 106\cdot 107^{2} + 29\cdot 107^{3} + 24\cdot 107^{4} +O(107^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 63 a + 59 + \left(94 a + 69\right)\cdot 107 + \left(90 a + 8\right)\cdot 107^{2} + \left(50 a + 94\right)\cdot 107^{3} + \left(71 a + 28\right)\cdot 107^{4} +O(107^{5})\)
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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 |
| $1$ | $1$ | $()$ | $9$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $40$ | $3$ | $(1,2,3)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.