Basic invariants
| Dimension: | $9$ |
| Group: | $S_6$ |
| Conductor: | \(14461155336531968\)\(\medspace = 2^{12} \cdot 15227^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.4.974528.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_{6}$ |
| Parity: | odd |
| Determinant: | 1.15227.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.4.974528.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - x^{4} + 4x^{3} - 4x^{2} + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 25 a + \left(2 a + 3\right)\cdot 47 + 46 a\cdot 47^{2} + \left(19 a + 5\right)\cdot 47^{3} + \left(29 a + 1\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 22 a + 3 + \left(44 a + 30\right)\cdot 47 + 42\cdot 47^{2} + \left(27 a + 45\right)\cdot 47^{3} + \left(17 a + 39\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 + 43\cdot 47 + 2\cdot 47^{2} + 8\cdot 47^{3} + 33\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 43 a + 34 + \left(18 a + 37\right)\cdot 47 + \left(30 a + 19\right)\cdot 47^{2} + \left(31 a + 4\right)\cdot 47^{3} + \left(35 a + 4\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 30 + 41\cdot 47 + 13\cdot 47^{2} + 40\cdot 47^{3} + 18\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a + 26 + \left(28 a + 32\right)\cdot 47 + \left(16 a + 14\right)\cdot 47^{2} + \left(15 a + 37\right)\cdot 47^{3} + \left(11 a + 43\right)\cdot 47^{4} +O(47^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $9$ | |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ | |
| $15$ | $2$ | $(1,2)$ | $3$ | ✓ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ | |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ | |
| $40$ | $3$ | $(1,2,3)$ | $0$ | |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ | |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ | |
| $144$ | $5$ | $(1,2,3,4,5)$ | $-1$ | |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ | |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |