Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(22291\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.22291.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | odd |
| Determinant: | 1.22291.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.0.22291.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + x^{4} - x^{3} - x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$:
\( x^{2} + 101x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 60 a + 91 + \left(85 a + 84\right)\cdot 113 + \left(109 a + 82\right)\cdot 113^{2} + \left(109 a + 88\right)\cdot 113^{3} + \left(39 a + 65\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 78 + 98\cdot 113 + 28\cdot 113^{2} + 66\cdot 113^{3} + 94\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 103 + 62\cdot 113 + 82\cdot 113^{2} + 56\cdot 113^{3} + 94\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 53 a + 20 + \left(27 a + 34\right)\cdot 113 + \left(3 a + 71\right)\cdot 113^{2} + \left(3 a + 55\right)\cdot 113^{3} + \left(73 a + 96\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 9 a + 26 + \left(47 a + 90\right)\cdot 113 + \left(31 a + 97\right)\cdot 113^{2} + \left(21 a + 36\right)\cdot 113^{3} + \left(69 a + 41\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 104 a + 21 + \left(65 a + 81\right)\cdot 113 + \left(81 a + 88\right)\cdot 113^{2} + \left(91 a + 34\right)\cdot 113^{3} + \left(43 a + 59\right)\cdot 113^{4} +O(113^{5})\)
|
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.