Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(16384000000000000\)\(\medspace = 2^{26} \cdot 5^{12} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.6400000.4 |
| 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 6.2.6400000.4 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 5x^{4} - 10x^{2} + 8x - 6 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 188 + 45\cdot 193 + 5\cdot 193^{2} + 143\cdot 193^{3} + 165\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 40 + 28\cdot 193 + 38\cdot 193^{2} + 152\cdot 193^{3} + 166\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 138 a + 69 + \left(177 a + 124\right)\cdot 193 + \left(10 a + 170\right)\cdot 193^{2} + \left(6 a + 11\right)\cdot 193^{3} + \left(5 a + 142\right)\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 154 + 160\cdot 193 + 14\cdot 193^{2} + 52\cdot 193^{3} + 88\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 55 a + 14 + \left(15 a + 164\right)\cdot 193 + \left(182 a + 3\right)\cdot 193^{2} + \left(186 a + 7\right)\cdot 193^{3} + \left(187 a + 141\right)\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 116 + 55\cdot 193 + 153\cdot 193^{2} + 19\cdot 193^{3} + 68\cdot 193^{4} +O(193^{5})\)
|
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.