Basic invariants
| Dimension: | $3$ |
| Group: | $A_4\times C_2$ |
| Conductor: | \(8829\)\(\medspace = 3^{4} \cdot 109 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.715149.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.109.2t1.a.a |
| Projective image: | $A_4$ |
| Projective stem field: | Galois closure of 4.0.962361.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{3} - 6x^{2} - 9x - 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$:
\( x^{2} + 69x + 7 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 64 a + 30 + \left(13 a + 34\right)\cdot 71 + \left(48 a + 57\right)\cdot 71^{2} + \left(15 a + 53\right)\cdot 71^{3} + \left(33 a + 56\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 a + 16 + \left(57 a + 69\right)\cdot 71 + \left(22 a + 68\right)\cdot 71^{2} + \left(55 a + 36\right)\cdot 71^{3} + \left(37 a + 36\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 68 a + 66 + \left(12 a + 16\right)\cdot 71 + \left(2 a + 14\right)\cdot 71^{2} + \left(14 a + 37\right)\cdot 71^{3} + \left(18 a + 70\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 69 + 64\cdot 71 + 11\cdot 71^{2} + 58\cdot 71^{3} + 58\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 43 + 52\cdot 71 + 54\cdot 71^{2} + 34\cdot 71^{3} + 39\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 a + 60 + \left(58 a + 45\right)\cdot 71 + \left(68 a + 5\right)\cdot 71^{2} + \left(56 a + 63\right)\cdot 71^{3} + \left(52 a + 21\right)\cdot 71^{4} +O(71^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-3$ |
| $3$ | $2$ | $(3,6)$ | $1$ |
| $3$ | $2$ | $(1,2)(3,6)$ | $-1$ |
| $4$ | $3$ | $(1,4,3)(2,5,6)$ | $0$ |
| $4$ | $3$ | $(1,3,4)(2,6,5)$ | $0$ |
| $4$ | $6$ | $(1,4,3,2,5,6)$ | $0$ |
| $4$ | $6$ | $(1,6,5,2,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.