Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(9227446944279201\)\(\medspace = 3^{16} \cdot 11^{8} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.8732691.3 |
| 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 6.0.8732691.3 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 3x^{4} - 2x^{3} + 6x^{2} + 9x + 5 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 31\cdot 47 + 30\cdot 47^{2} + 28\cdot 47^{3} + 34\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 41 a + 8 + \left(41 a + 26\right)\cdot 47 + \left(15 a + 42\right)\cdot 47^{2} + \left(3 a + 41\right)\cdot 47^{3} + \left(10 a + 44\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 a + 43 + \left(5 a + 21\right)\cdot 47 + \left(31 a + 32\right)\cdot 47^{2} + \left(43 a + 32\right)\cdot 47^{3} + \left(36 a + 14\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 42 + 31\cdot 47 + 30\cdot 47^{2} + 38\cdot 47^{3} + 45\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 27 a + 40 + \left(2 a + 25\right)\cdot 47 + \left(12 a + 38\right)\cdot 47^{2} + \left(4 a + 24\right)\cdot 47^{3} + \left(13 a + 36\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 20 a + \left(44 a + 4\right)\cdot 47 + \left(34 a + 13\right)\cdot 47^{2} + \left(42 a + 21\right)\cdot 47^{3} + \left(33 a + 11\right)\cdot 47^{4} +O(47^{5})\)
|
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.