Basic invariants
| Dimension: | $9$ |
| Group: | $S_6$ |
| Conductor: | \(106\!\cdots\!789\)\(\medspace = 3^{9} \cdot 175327^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.6.4733829.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_{6}$ |
| Parity: | even |
| Determinant: | 1.525981.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.6.4733829.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 6x^{4} + 4x^{3} + 9x^{2} - 3x - 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 22 a + 2 + \left(12 a + 12\right)\cdot 31 + \left(28 a + 27\right)\cdot 31^{2} + \left(25 a + 17\right)\cdot 31^{3} + \left(14 a + 12\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 28 + 9\cdot 31 + 20\cdot 31^{2} + 14\cdot 31^{3} + 15\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 a + 15 + \left(18 a + 15\right)\cdot 31 + \left(2 a + 9\right)\cdot 31^{2} + \left(5 a + 10\right)\cdot 31^{3} + \left(16 a + 16\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 + 25\cdot 31 + 26\cdot 31^{2} + 21\cdot 31^{3} + 26\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 27 a + 10 + \left(16 a + 27\right)\cdot 31 + 27\cdot 31^{2} + \left(23 a + 6\right)\cdot 31^{3} + \left(8 a + 29\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a + 2 + \left(14 a + 3\right)\cdot 31 + \left(30 a + 12\right)\cdot 31^{2} + \left(7 a + 21\right)\cdot 31^{3} + \left(22 a + 23\right)\cdot 31^{4} +O(31^{5})\)
|
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.