Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(637\!\cdots\!321\)\(\medspace = 92779^{6} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.4.92779.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 30T164 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.4.92779.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 2x^{4} + 3x^{3} - x^{2} - 2x + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 92 a + 45 + \left(107 a + 114\right)\cdot 193 + \left(54 a + 103\right)\cdot 193^{2} + \left(109 a + 89\right)\cdot 193^{3} + \left(102 a + 158\right)\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 102 a + 136 + \left(59 a + 130\right)\cdot 193 + \left(87 a + 96\right)\cdot 193^{2} + \left(18 a + 157\right)\cdot 193^{3} + \left(77 a + 10\right)\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 91 a + 45 + \left(133 a + 88\right)\cdot 193 + \left(105 a + 124\right)\cdot 193^{2} + \left(174 a + 88\right)\cdot 193^{3} + \left(115 a + 69\right)\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 101 + 172\cdot 193 + 32\cdot 193^{2} + 49\cdot 193^{3} + 167\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 101 a + 137 + \left(85 a + 129\right)\cdot 193 + \left(138 a + 50\right)\cdot 193^{2} + \left(83 a + 144\right)\cdot 193^{3} + \left(90 a + 151\right)\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 116 + 136\cdot 193 + 170\cdot 193^{2} + 49\cdot 193^{3} + 21\cdot 193^{4} +O(193^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $10$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $15$ | $2$ | $(1,2)$ | $-2$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.