Basic invariants
| Dimension: | $3$ |
| Group: | $A_4\times C_2$ |
| Conductor: | \(7209\)\(\medspace = 3^{4} \cdot 89 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.583929.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.89.2t1.a.a |
| Projective image: | $A_4$ |
| Projective stem field: | Galois closure of 4.0.641601.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 5x^{3} - 3x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{2} + 16x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 a + 3 + \left(16 a + 15\right)\cdot 17 + \left(7 a + 13\right)\cdot 17^{2} + \left(16 a + 13\right)\cdot 17^{3} + \left(12 a + 7\right)\cdot 17^{4} + \left(12 a + 2\right)\cdot 17^{5} + \left(8 a + 7\right)\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 + 2\cdot 17 + 15\cdot 17^{2} + 15\cdot 17^{3} + 14\cdot 17^{4} + 14\cdot 17^{5} + 8\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 a + 11 + 6\cdot 17 + \left(9 a + 5\right)\cdot 17^{2} + 5\cdot 17^{3} + \left(4 a + 4\right)\cdot 17^{4} + \left(4 a + 2\right)\cdot 17^{5} + \left(8 a + 3\right)\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 a + \left(15 a + 15\right)\cdot 17 + \left(5 a + 1\right)\cdot 17^{2} + \left(3 a + 14\right)\cdot 17^{3} + \left(a + 2\right)\cdot 17^{4} + \left(16 a + 9\right)\cdot 17^{5} + \left(14 a + 2\right)\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a + 7 + \left(a + 6\right)\cdot 17 + \left(11 a + 9\right)\cdot 17^{2} + \left(13 a + 11\right)\cdot 17^{3} + 15 a\cdot 17^{4} + 7\cdot 17^{5} + \left(2 a + 1\right)\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 2 + 5\cdot 17 + 5\cdot 17^{2} + 7\cdot 17^{3} + 3\cdot 17^{4} + 15\cdot 17^{5} + 10\cdot 17^{6} +O(17^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,3)(2,6)(4,5)$ | $-3$ |
| $3$ | $2$ | $(1,3)$ | $1$ |
| $3$ | $2$ | $(1,3)(2,6)$ | $-1$ |
| $4$ | $3$ | $(1,2,4)(3,6,5)$ | $0$ |
| $4$ | $3$ | $(1,4,2)(3,5,6)$ | $0$ |
| $4$ | $6$ | $(1,6,5,3,2,4)$ | $0$ |
| $4$ | $6$ | $(1,4,2,3,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.