Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(22784\)\(\medspace = 2^{8} \cdot 89 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.182272.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.89.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.2.182272.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + x^{4} - 3x^{2} + 2x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$:
\( x^{2} + 70x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 26 a + 61 + \left(42 a + 9\right)\cdot 73 + 46 a\cdot 73^{2} + \left(51 a + 27\right)\cdot 73^{3} + \left(7 a + 64\right)\cdot 73^{4} +O(73^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 4 + 55\cdot 73 + 22\cdot 73^{2} + 55\cdot 73^{4} +O(73^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 47 a + 66 + \left(30 a + 37\right)\cdot 73 + \left(26 a + 24\right)\cdot 73^{2} + \left(21 a + 62\right)\cdot 73^{3} + \left(65 a + 35\right)\cdot 73^{4} +O(73^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 22 a + 18 + \left(25 a + 72\right)\cdot 73 + 73^{2} + \left(48 a + 42\right)\cdot 73^{3} + \left(39 a + 24\right)\cdot 73^{4} +O(73^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 61 + 63\cdot 73 + 45\cdot 73^{2} + 47\cdot 73^{3} + 16\cdot 73^{4} +O(73^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 51 a + 11 + \left(47 a + 53\right)\cdot 73 + \left(72 a + 50\right)\cdot 73^{2} + \left(24 a + 39\right)\cdot 73^{3} + \left(33 a + 22\right)\cdot 73^{4} +O(73^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(3,5)$ | $2$ |
| $9$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,3,5)(2,4,6)$ | $-2$ |
| $4$ | $3$ | $(1,3,5)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,4,3,6,5,2)$ | $0$ |
| $12$ | $6$ | $(2,4,6)(3,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.