Basic invariants
| Dimension: | $3$ |
| Group: | $A_4\times C_2$ |
| Conductor: | \(17101\)\(\medspace = 7^{2} \cdot 349 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.837949.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.349.2t1.a.a |
| Projective image: | $A_4$ |
| Projective stem field: | Galois closure of 4.0.5968249.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 3x^{4} - 9x^{3} + 4x^{2} - 9x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 24 a + \left(13 a + 4\right)\cdot 41 + \left(24 a + 26\right)\cdot 41^{2} + \left(34 a + 2\right)\cdot 41^{3} + \left(28 a + 36\right)\cdot 41^{4} + \left(30 a + 15\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 a + 31 + \left(27 a + 20\right)\cdot 41 + \left(16 a + 3\right)\cdot 41^{2} + 6 a\cdot 41^{3} + \left(12 a + 6\right)\cdot 41^{4} + \left(10 a + 38\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 + 31\cdot 41 + 28\cdot 41^{2} + 18\cdot 41^{3} + 34\cdot 41^{4} + 38\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 a + 23 + \left(11 a + 22\right)\cdot 41 + \left(12 a + 35\right)\cdot 41^{2} + \left(28 a + 30\right)\cdot 41^{3} + \left(3 a + 13\right)\cdot 41^{4} + \left(4 a + 29\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 38 + 39\cdot 41 + 8\cdot 41^{2} + 8\cdot 41^{3} + 36\cdot 41^{4} + 3\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 30 a + 15 + \left(29 a + 4\right)\cdot 41 + \left(28 a + 20\right)\cdot 41^{2} + \left(12 a + 21\right)\cdot 41^{3} + \left(37 a + 37\right)\cdot 41^{4} + \left(36 a + 37\right)\cdot 41^{5} +O(41^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-3$ |
| $3$ | $2$ | $(1,2)$ | $1$ |
| $3$ | $2$ | $(1,2)(3,5)$ | $-1$ |
| $4$ | $3$ | $(1,3,4)(2,5,6)$ | $0$ |
| $4$ | $3$ | $(1,4,3)(2,6,5)$ | $0$ |
| $4$ | $6$ | $(1,5,6,2,3,4)$ | $0$ |
| $4$ | $6$ | $(1,4,3,2,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.