Basic invariants
| Dimension: | $3$ |
| Group: | $A_4\times C_2$ |
| Conductor: | \(18571\)\(\medspace = 7^{2} \cdot 379 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.909979.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.379.2t1.a.a |
| Projective image: | $A_4$ |
| Projective stem field: | Galois closure of 4.4.7038409.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 7x^{4} - 12x^{3} + 21x^{2} - 15x + 13 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 17 a + 23 + \left(17 a + 38\right)\cdot 41 + \left(36 a + 13\right)\cdot 41^{2} + \left(21 a + 19\right)\cdot 41^{3} + \left(11 a + 8\right)\cdot 41^{4} + \left(26 a + 8\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 a + 33 + \left(23 a + 32\right)\cdot 41 + \left(4 a + 23\right)\cdot 41^{2} + \left(19 a + 7\right)\cdot 41^{3} + \left(29 a + 21\right)\cdot 41^{4} + \left(14 a + 34\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 39 a + 39 + \left(3 a + 25\right)\cdot 41 + \left(23 a + 2\right)\cdot 41^{2} + \left(37 a + 18\right)\cdot 41^{3} + \left(33 a + 9\right)\cdot 41^{4} + \left(17 a + 17\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 23 + 31\cdot 41 + 7\cdot 41^{2} + 36\cdot 41^{3} + 25\cdot 41^{4} + 18\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 15 + 36\cdot 41 + 6\cdot 41^{2} + 16\cdot 41^{3} + 25\cdot 41^{4} + 7\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 2 a + 33 + \left(37 a + 39\right)\cdot 41 + \left(17 a + 26\right)\cdot 41^{2} + \left(3 a + 25\right)\cdot 41^{3} + \left(7 a + 32\right)\cdot 41^{4} + \left(23 a + 36\right)\cdot 41^{5} +O(41^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-3$ |
| $3$ | $2$ | $(3,6)$ | $1$ |
| $3$ | $2$ | $(1,2)(3,6)$ | $-1$ |
| $4$ | $3$ | $(1,4,3)(2,5,6)$ | $0$ |
| $4$ | $3$ | $(1,3,4)(2,6,5)$ | $0$ |
| $4$ | $6$ | $(1,4,3,2,5,6)$ | $0$ |
| $4$ | $6$ | $(1,6,5,2,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.