Basic invariants
| Dimension: | $5$ |
| Group: | $A_6$ |
| Conductor: | \(6400000000\)\(\medspace = 2^{14} \cdot 5^{8} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.6400000000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_6$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_6$ |
| Projective stem field: | Galois closure of 6.2.6400000000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 15x^{4} + 50x^{2} - 4x - 82 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 a + 16 + \left(19 a + 17\right)\cdot 23 + \left(a + 19\right)\cdot 23^{2} + \left(13 a + 5\right)\cdot 23^{3} + 12\cdot 23^{4} + \left(9 a + 1\right)\cdot 23^{5} + \left(11 a + 17\right)\cdot 23^{6} + \left(18 a + 18\right)\cdot 23^{7} +O(23^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 + 16\cdot 23 + 15\cdot 23^{2} + 15\cdot 23^{3} + 3\cdot 23^{4} + 14\cdot 23^{5} + 19\cdot 23^{6} + 23^{7} +O(23^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 a + 6 + \left(12 a + 8\right)\cdot 23 + \left(11 a + 7\right)\cdot 23^{2} + \left(6 a + 14\right)\cdot 23^{3} + \left(13 a + 7\right)\cdot 23^{4} + \left(5 a + 5\right)\cdot 23^{5} + 18 a\cdot 23^{6} + \left(19 a + 10\right)\cdot 23^{7} +O(23^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 a + 22 + \left(3 a + 6\right)\cdot 23 + \left(21 a + 4\right)\cdot 23^{2} + \left(9 a + 7\right)\cdot 23^{3} + 22 a\cdot 23^{4} + \left(13 a + 19\right)\cdot 23^{5} + \left(11 a + 7\right)\cdot 23^{6} + \left(4 a + 21\right)\cdot 23^{7} +O(23^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 19 + 16\cdot 23 + 3\cdot 23^{2} + 10\cdot 23^{3} + 17\cdot 23^{4} + 2\cdot 23^{5} + 16\cdot 23^{6} + 8\cdot 23^{7} +O(23^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 16 a + 20 + \left(10 a + 2\right)\cdot 23 + \left(11 a + 18\right)\cdot 23^{2} + \left(16 a + 15\right)\cdot 23^{3} + \left(9 a + 4\right)\cdot 23^{4} + \left(17 a + 3\right)\cdot 23^{5} + \left(4 a + 8\right)\cdot 23^{6} + \left(3 a + 8\right)\cdot 23^{7} +O(23^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.