Basic invariants
| Dimension: | $5$ |
| Group: | $A_6$ |
| Conductor: | \(700569\)\(\medspace = 3^{6} \cdot 31^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.700569.1 |
| 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 6.2.700569.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{3} - 3x^{2} - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$:
\( x^{2} + 97x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 95 + 81\cdot 101 + 30\cdot 101^{2} + 62\cdot 101^{3} + 50\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 66 a + 59 + \left(45 a + 55\right)\cdot 101 + \left(47 a + 81\right)\cdot 101^{2} + \left(83 a + 27\right)\cdot 101^{3} + \left(37 a + 55\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 a + 56 + \left(40 a + 89\right)\cdot 101 + \left(83 a + 13\right)\cdot 101^{2} + \left(26 a + 76\right)\cdot 101^{3} + 41\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 82 a + 31 + \left(60 a + 29\right)\cdot 101 + \left(17 a + 4\right)\cdot 101^{2} + \left(74 a + 100\right)\cdot 101^{3} + \left(100 a + 15\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 42 + 76\cdot 101 + 47\cdot 101^{2} + 25\cdot 101^{3} + 16\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 35 a + 20 + \left(55 a + 71\right)\cdot 101 + \left(53 a + 23\right)\cdot 101^{2} + \left(17 a + 11\right)\cdot 101^{3} + \left(63 a + 22\right)\cdot 101^{4} +O(101^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.