Basic invariants
| Dimension: | $3$ |
| Group: | $A_4\times C_2$ |
| Conductor: | \(18473\)\(\medspace = 7^{2} \cdot 13 \cdot 29 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.6.905177.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.377.2t1.a.a |
| Projective image: | $A_4$ |
| Projective stem field: | Galois closure of 4.4.6964321.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 7x^{4} + 9x^{3} + 7x^{2} - 9x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 a + 36 + \left(19 a + 23\right)\cdot 41 + \left(36 a + 1\right)\cdot 41^{2} + \left(26 a + 38\right)\cdot 41^{3} + \left(17 a + 6\right)\cdot 41^{4} + \left(28 a + 21\right)\cdot 41^{5} + \left(17 a + 18\right)\cdot 41^{6} +O(41^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 + 6\cdot 41 + 14\cdot 41^{2} + 22\cdot 41^{3} + 26\cdot 41^{4} + 2\cdot 41^{5} + 7\cdot 41^{6} +O(41^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 16 a + 10 + \left(25 a + 16\right)\cdot 41 + 13\cdot 41^{2} + \left(38 a + 32\right)\cdot 41^{3} + \left(19 a + 35\right)\cdot 41^{4} + \left(31 a + 2\right)\cdot 41^{5} + \left(13 a + 17\right)\cdot 41^{6} +O(41^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 + 7\cdot 41 + 12\cdot 41^{2} + 7\cdot 41^{3} + 4\cdot 41^{4} + 12\cdot 41^{5} + 10\cdot 41^{6} +O(41^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 34 a + 16 + \left(21 a + 33\right)\cdot 41 + \left(4 a + 9\right)\cdot 41^{2} + 14 a\cdot 41^{3} + \left(23 a + 33\right)\cdot 41^{4} + \left(12 a + 6\right)\cdot 41^{5} + \left(23 a + 2\right)\cdot 41^{6} +O(41^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 25 a + 17 + \left(15 a + 35\right)\cdot 41 + \left(40 a + 30\right)\cdot 41^{2} + \left(2 a + 22\right)\cdot 41^{3} + \left(21 a + 16\right)\cdot 41^{4} + \left(9 a + 36\right)\cdot 41^{5} + \left(27 a + 26\right)\cdot 41^{6} +O(41^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-3$ |
| $3$ | $2$ | $(1,5)$ | $1$ |
| $3$ | $2$ | $(1,5)(2,4)$ | $-1$ |
| $4$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $4$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
| $4$ | $6$ | $(1,4,6,5,2,3)$ | $0$ |
| $4$ | $6$ | $(1,3,2,5,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.