Basic invariants
| Dimension: | $3$ |
| Group: | $A_4\times C_2$ |
| Conductor: | \(13769\)\(\medspace = 7^{2} \cdot 281 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.674681.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.281.2t1.a.a |
| Projective image: | $A_4$ |
| Projective stem field: | Galois closure of 4.0.3869089.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + x^{4} + 3x^{3} - 4x^{2} - 4x - 8 \)
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The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 a + 11 + 4 a\cdot 29 + \left(6 a + 8\right)\cdot 29^{2} + \left(20 a + 18\right)\cdot 29^{3} + \left(24 a + 21\right)\cdot 29^{4} + \left(3 a + 6\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 + 22\cdot 29 + 17\cdot 29^{2} + 2\cdot 29^{3} + 5\cdot 29^{4} + 26\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 25 a + 5 + \left(22 a + 22\right)\cdot 29 + \left(2 a + 18\right)\cdot 29^{2} + \left(4 a + 12\right)\cdot 29^{3} + \left(22 a + 17\right)\cdot 29^{4} + \left(16 a + 11\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 + 3\cdot 29 + 27\cdot 29^{2} + 25\cdot 29^{3} + 25\cdot 29^{4} + 25\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 4 a + 14 + \left(6 a + 24\right)\cdot 29 + \left(26 a + 9\right)\cdot 29^{2} + \left(24 a + 1\right)\cdot 29^{3} + \left(6 a + 8\right)\cdot 29^{4} + \left(12 a + 15\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 21 a + 22 + \left(24 a + 13\right)\cdot 29 + \left(22 a + 5\right)\cdot 29^{2} + \left(8 a + 26\right)\cdot 29^{3} + \left(4 a + 8\right)\cdot 29^{4} + \left(25 a + 1\right)\cdot 29^{5} +O(29^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,4)(3,5)$ | $-3$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $3$ | $2$ | $(1,6)(2,4)$ | $-1$ |
| $4$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $4$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $4$ | $6$ | $(1,5,4,6,3,2)$ | $0$ |
| $4$ | $6$ | $(1,2,3,6,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.