Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(5648\)\(\medspace = 2^{4} \cdot 353 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.22592.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.353.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.22592.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 2x^{4} - x^{2} + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 2 + 23\cdot 29 + 28\cdot 29^{2} + 20\cdot 29^{3} + 22\cdot 29^{4} +O(29^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 15 a + 14 + \left(17 a + 9\right)\cdot 29 + 29^{2} + \left(22 a + 21\right)\cdot 29^{3} + \left(5 a + 19\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 18 a + 7 + \left(22 a + 14\right)\cdot 29 + \left(22 a + 20\right)\cdot 29^{2} + \left(10 a + 10\right)\cdot 29^{3} + \left(21 a + 23\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 14 a + 2 + \left(11 a + 24\right)\cdot 29 + \left(28 a + 15\right)\cdot 29^{2} + \left(6 a + 14\right)\cdot 29^{3} + \left(23 a + 26\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 25 + 22\cdot 29 + 24\cdot 29^{2} + 6\cdot 29^{3} + 20\cdot 29^{4} +O(29^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 11 a + 10 + \left(6 a + 22\right)\cdot 29 + \left(6 a + 24\right)\cdot 29^{2} + \left(18 a + 12\right)\cdot 29^{3} + \left(7 a + 3\right)\cdot 29^{4} +O(29^{5})\)
| |
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 value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(2,4)$ | $2$ |
| $9$ | $2$ | $(2,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,2,4)$ | $1$ |
| $4$ | $3$ | $(1,2,4)(3,5,6)$ | $-2$ |
| $18$ | $4$ | $(1,3)(2,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,5,2,6,4,3)$ | $0$ |
| $12$ | $6$ | $(2,4)(3,5,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.