Basic invariants
| Dimension: | $3$ |
| Group: | $A_4\times C_2$ |
| Conductor: | \(10927\)\(\medspace = 7^{2} \cdot 223 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.4.535423.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.223.2t1.a.a |
| Projective image: | $A_4$ |
| Projective stem field: | Galois closure of 4.0.2436721.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - 2x^{4} + 3x^{2} + 6x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 22 a + 7 + \left(14 a + 16\right)\cdot 29 + \left(10 a + 3\right)\cdot 29^{2} + \left(14 a + 27\right)\cdot 29^{3} + \left(26 a + 28\right)\cdot 29^{4} + \left(2 a + 2\right)\cdot 29^{5} + \left(3 a + 4\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( a + 14 + \left(15 a + 13\right)\cdot 29 + \left(18 a + 11\right)\cdot 29^{2} + \left(15 a + 21\right)\cdot 29^{3} + \left(18 a + 5\right)\cdot 29^{4} + \left(3 a + 4\right)\cdot 29^{5} + \left(24 a + 11\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 a + 1 + \left(14 a + 10\right)\cdot 29 + \left(18 a + 12\right)\cdot 29^{2} + \left(14 a + 1\right)\cdot 29^{3} + \left(2 a + 2\right)\cdot 29^{4} + \left(26 a + 20\right)\cdot 29^{5} + \left(25 a + 16\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 28 a + 19 + 13 a\cdot 29 + \left(10 a + 2\right)\cdot 29^{2} + \left(13 a + 23\right)\cdot 29^{3} + \left(10 a + 24\right)\cdot 29^{4} + \left(25 a + 3\right)\cdot 29^{5} + \left(4 a + 12\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 9 + 3\cdot 29 + 4\cdot 29^{2} + 21\cdot 29^{3} + 10\cdot 29^{4} + 24\cdot 29^{5} + 9\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 + 14\cdot 29 + 24\cdot 29^{2} + 21\cdot 29^{3} + 14\cdot 29^{4} + 2\cdot 29^{5} + 4\cdot 29^{6} +O(29^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,3)(2,4)(5,6)$ | $-3$ |
| $3$ | $2$ | $(1,3)$ | $1$ |
| $3$ | $2$ | $(1,3)(2,4)$ | $-1$ |
| $4$ | $3$ | $(1,2,5)(3,4,6)$ | $0$ |
| $4$ | $3$ | $(1,5,2)(3,6,4)$ | $0$ |
| $4$ | $6$ | $(1,4,6,3,2,5)$ | $0$ |
| $4$ | $6$ | $(1,5,2,3,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.