Basic invariants
| Dimension: | $9$ |
| Group: | $S_6$ |
| Conductor: | \(655360000000000\)\(\medspace = 2^{26} \cdot 5^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.6400000.3 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 20T145 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.2.6400000.3 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 5x^{4} - 5x^{2} + 10x - 5 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$:
\( x^{2} + 192x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 33 a + 17 + \left(181 a + 9\right)\cdot 193 + \left(57 a + 51\right)\cdot 193^{2} + \left(26 a + 151\right)\cdot 193^{3} + \left(159 a + 52\right)\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 161 a + 78 + \left(5 a + 112\right)\cdot 193 + \left(5 a + 104\right)\cdot 193^{2} + \left(68 a + 160\right)\cdot 193^{3} + \left(143 a + 192\right)\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 142 a + 124 + \left(91 a + 3\right)\cdot 193 + \left(145 a + 169\right)\cdot 193^{2} + \left(15 a + 26\right)\cdot 193^{3} + \left(74 a + 7\right)\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 160 a + 50 + \left(11 a + 157\right)\cdot 193 + \left(135 a + 120\right)\cdot 193^{2} + \left(166 a + 119\right)\cdot 193^{3} + \left(33 a + 185\right)\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 32 a + 46 + \left(187 a + 150\right)\cdot 193 + \left(187 a + 103\right)\cdot 193^{2} + \left(124 a + 30\right)\cdot 193^{3} + \left(49 a + 75\right)\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 51 a + 73 + \left(101 a + 146\right)\cdot 193 + \left(47 a + 29\right)\cdot 193^{2} + \left(177 a + 90\right)\cdot 193^{3} + \left(118 a + 65\right)\cdot 193^{4} +O(193^{5})\)
|
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.