Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(43531\)\(\medspace = 101 \cdot 431 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.43531.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | odd |
| Determinant: | 1.43531.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.0.43531.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 157 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 157 }$:
\( x^{2} + 152x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 119 + 95\cdot 157 + 44\cdot 157^{2} + 72\cdot 157^{3} + 99\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 148 a + 40 + \left(95 a + 38\right)\cdot 157 + \left(94 a + 121\right)\cdot 157^{2} + \left(106 a + 72\right)\cdot 157^{3} + \left(131 a + 97\right)\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 59 + 63\cdot 157 + 5\cdot 157^{2} + 131\cdot 157^{3} + 46\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 25 a + 145 + \left(88 a + 57\right)\cdot 157 + \left(107 a + 68\right)\cdot 157^{2} + \left(82 a + 81\right)\cdot 157^{3} + \left(114 a + 93\right)\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 9 a + 152 + \left(61 a + 55\right)\cdot 157 + \left(62 a + 27\right)\cdot 157^{2} + \left(50 a + 40\right)\cdot 157^{3} + \left(25 a + 21\right)\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 132 a + 113 + \left(68 a + 2\right)\cdot 157 + \left(49 a + 47\right)\cdot 157^{2} + \left(74 a + 73\right)\cdot 157^{3} + \left(42 a + 112\right)\cdot 157^{4} +O(157^{5})\)
|
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.