Basic invariants
Dimension: | $3$ |
Group: | $A_4\times C_2$ |
Conductor: | \(8183\)\(\medspace = 7^{2} \cdot 167 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 6.0.400967.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $A_4\times C_2$ |
Parity: | odd |
Projective image: | $A_4$ |
Projective field: | Galois closure of 4.4.1366561.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 3 a + 9 + \left(17 a + 9\right)\cdot 29 + \left(27 a + 4\right)\cdot 29^{2} + 25 a\cdot 29^{3} + \left(9 a + 18\right)\cdot 29^{4} +O(29^{5})\)
$r_{ 2 }$ |
$=$ |
\( 4 + 20\cdot 29 + 26\cdot 29^{2} + 24\cdot 29^{3} + 25\cdot 29^{4} +O(29^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 18 a + 17 + \left(25 a + 1\right)\cdot 29 + \left(3 a + 11\right)\cdot 29^{2} + 12 a\cdot 29^{3} + \left(11 a + 22\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 11 a + 20 + \left(3 a + 24\right)\cdot 29 + \left(25 a + 4\right)\cdot 29^{2} + \left(16 a + 28\right)\cdot 29^{3} + \left(17 a + 8\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 15 + 26\cdot 29 + 29^{2} + 18\cdot 29^{3} + 28\cdot 29^{4} +O(29^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 26 a + 24 + \left(11 a + 4\right)\cdot 29 + \left(a + 9\right)\cdot 29^{2} + \left(3 a + 15\right)\cdot 29^{3} + \left(19 a + 12\right)\cdot 29^{4} +O(29^{5})\)
| |
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 values |
$c1$ | |||
$1$ | $1$ | $()$ | $3$ |
$1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
$3$ | $2$ | $(3,4)$ | $1$ |
$3$ | $2$ | $(1,6)(3,4)$ | $-1$ |
$4$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
$4$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
$4$ | $6$ | $(1,2,3,6,5,4)$ | $0$ |
$4$ | $6$ | $(1,4,5,6,3,2)$ | $0$ |