Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(18688\)\(\medspace = 2^{8} \cdot 73 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.149504.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.73.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.2.149504.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + x^{4} - x^{2} + 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$:
\( x^{2} + 82x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 54 a + 81 + \left(10 a + 84\right)\cdot 89 + \left(25 a + 80\right)\cdot 89^{2} + \left(31 a + 68\right)\cdot 89^{3} + \left(53 a + 51\right)\cdot 89^{4} +O(89^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 59 + 53\cdot 89 + 78\cdot 89^{2} + 63\cdot 89^{3} + 62\cdot 89^{4} +O(89^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 83 + 27\cdot 89 + 81\cdot 89^{2} + 33\cdot 89^{3} + 26\cdot 89^{4} +O(89^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 64 a + 59 + \left(62 a + 32\right)\cdot 89 + \left(34 a + 22\right)\cdot 89^{2} + \left(45 a + 44\right)\cdot 89^{3} + \left(80 a + 7\right)\cdot 89^{4} +O(89^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 35 a + 14 + \left(78 a + 16\right)\cdot 89 + \left(63 a + 68\right)\cdot 89^{2} + \left(57 a + 84\right)\cdot 89^{3} + \left(35 a + 37\right)\cdot 89^{4} +O(89^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 25 a + 62 + \left(26 a + 51\right)\cdot 89 + \left(54 a + 24\right)\cdot 89^{2} + \left(43 a + 60\right)\cdot 89^{3} + \left(8 a + 80\right)\cdot 89^{4} +O(89^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(2,5)$ | $2$ |
| $9$ | $2$ | $(2,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,2,5)$ | $1$ |
| $4$ | $3$ | $(1,2,5)(3,4,6)$ | $-2$ |
| $18$ | $4$ | $(1,3)(2,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,4,2,6,5,3)$ | $0$ |
| $12$ | $6$ | $(2,5)(3,4,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.