Basic invariants
| Dimension: | $3$ |
| Group: | $A_4\times C_2$ |
| Conductor: | \(11711\)\(\medspace = 7^{2} \cdot 239 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.573839.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.239.2t1.a.a |
| Projective image: | $A_4$ |
| Projective stem field: | Galois closure of 4.4.2798929.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 4x^{4} - 3x^{3} + 8x^{2} - 4x + 8 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 a + 10 + \left(5 a + 8\right)\cdot 29 + \left(28 a + 11\right)\cdot 29^{2} + \left(10 a + 1\right)\cdot 29^{3} + \left(2 a + 13\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 a + 26 + \left(6 a + 25\right)\cdot 29 + \left(18 a + 15\right)\cdot 29^{2} + \left(6 a + 14\right)\cdot 29^{3} + \left(22 a + 7\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 28 + 13\cdot 29 + 27\cdot 29^{2} + 19\cdot 29^{3} + 13\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 27 + 17\cdot 29^{2} + 22\cdot 29^{3} + 13\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 14 a + 14 + \left(22 a + 14\right)\cdot 29 + \left(10 a + 13\right)\cdot 29^{2} + 22 a\cdot 29^{3} + \left(6 a + 25\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 17 a + 12 + \left(23 a + 23\right)\cdot 29 + 29^{2} + \left(18 a + 28\right)\cdot 29^{3} + \left(26 a + 13\right)\cdot 29^{4} +O(29^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(3,4)$ | $1$ |
| $3$ | $2$ | $(1,6)(3,4)$ | $-1$ |
| $4$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $4$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
| $4$ | $6$ | $(1,2,3,6,5,4)$ | $0$ |
| $4$ | $6$ | $(1,4,5,6,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.