Basic invariants
| Dimension: | $8$ |
| Group: | $A_6$ |
| Conductor: | \(20390869140625\)\(\medspace = 5^{12} \cdot 17^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.112890625.1 |
| 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 6.2.112890625.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 5x^{2} - 11x - 13 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 16 a + 29 + \left(29 a + 4\right)\cdot 47 + \left(12 a + 2\right)\cdot 47^{2} + \left(21 a + 23\right)\cdot 47^{3} + \left(29 a + 37\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 31 + 33\cdot 47 + 4\cdot 47^{2} + 34\cdot 47^{3} + 24\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 31 a + 14 + 17 a\cdot 47 + \left(34 a + 45\right)\cdot 47^{2} + \left(25 a + 5\right)\cdot 47^{3} + \left(17 a + 28\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 a + 41 + \left(10 a + 26\right)\cdot 47 + \left(42 a + 1\right)\cdot 47^{2} + \left(13 a + 20\right)\cdot 47^{3} + \left(23 a + 45\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 38 a + 12 + \left(36 a + 38\right)\cdot 47 + \left(4 a + 28\right)\cdot 47^{2} + \left(33 a + 5\right)\cdot 47^{3} + \left(23 a + 31\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 16 + 37\cdot 47 + 11\cdot 47^{2} + 5\cdot 47^{3} + 21\cdot 47^{4} +O(47^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $8$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.