Basic invariants
| Dimension: | $8$ |
| Group: | $A_6$ |
| Conductor: | \(749\!\cdots\!881\)\(\medspace = 3^{16} \cdot 7^{6} \cdot 23^{6} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.6.170067681.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $A_6$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_6$ |
| Projective stem field: | Galois closure of 6.6.170067681.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} - 9x^{4} + 14x^{3} + 27x^{2} - 3x - 10 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 27 + 19\cdot 31 + 20\cdot 31^{2} + 2\cdot 31^{3} + 10\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 a + 29 + \left(23 a + 28\right)\cdot 31 + 11 a\cdot 31^{2} + \left(23 a + 12\right)\cdot 31^{3} + 30\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 + 5\cdot 31 + 29\cdot 31^{2} + 20\cdot 31^{3} + 24\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 12 a + 13 + \left(14 a + 29\right)\cdot 31 + \left(17 a + 25\right)\cdot 31^{2} + \left(13 a + 15\right)\cdot 31^{3} + \left(29 a + 2\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 19 a + 6 + \left(16 a + 15\right)\cdot 31 + \left(13 a + 15\right)\cdot 31^{2} + \left(17 a + 25\right)\cdot 31^{3} + \left(a + 16\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 11 a + 7 + \left(7 a + 25\right)\cdot 31 + 19 a\cdot 31^{2} + \left(7 a + 16\right)\cdot 31^{3} + \left(30 a + 8\right)\cdot 31^{4} +O(31^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $8$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.