Basic invariants
| Dimension: | $9$ |
| Group: | $S_6$ |
| Conductor: | \(196073702927424\)\(\medspace = 2^{6} \cdot 3^{12} \cdot 7^{8} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.12446784.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_{6}$ |
| Parity: | odd |
| Determinant: | 1.4.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.0.12446784.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 5x^{4} - 2x^{3} + 9x^{2} - 8x + 16 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 90 a + 84 + \left(66 a + 4\right)\cdot 157 + \left(81 a + 121\right)\cdot 157^{2} + \left(156 a + 37\right)\cdot 157^{3} + \left(15 a + 98\right)\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( a + 124 + \left(3 a + 52\right)\cdot 157 + \left(154 a + 33\right)\cdot 157^{2} + \left(62 a + 31\right)\cdot 157^{3} + \left(140 a + 143\right)\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 67 a + 63 + \left(90 a + 90\right)\cdot 157 + \left(75 a + 147\right)\cdot 157^{2} + 110\cdot 157^{3} + \left(141 a + 21\right)\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 156 a + 129 + \left(153 a + 66\right)\cdot 157 + \left(2 a + 15\right)\cdot 157^{2} + \left(94 a + 35\right)\cdot 157^{3} + \left(16 a + 154\right)\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 43 + 58\cdot 157 + 29\cdot 157^{2} + 157^{3} + 61\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 28 + 41\cdot 157 + 124\cdot 157^{2} + 97\cdot 157^{3} + 149\cdot 157^{4} +O(157^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $9$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $40$ | $3$ | $(1,2,3)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.