Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(24489\)\(\medspace = 3^{3} \cdot 907 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.73467.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.2721.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.73467.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 3x^{4} - 4x^{3} + 4x^{2} - 3x + 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 a + 5 + \left(5 a + 7\right)\cdot 19 + \left(3 a + 9\right)\cdot 19^{2} + \left(3 a + 1\right)\cdot 19^{3} + \left(9 a + 16\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 + 14\cdot 19 + 13\cdot 19^{2} + 15\cdot 19^{3} + 17\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 a + \left(14 a + 18\right)\cdot 19 + \left(14 a + 16\right)\cdot 19^{2} + \left(8 a + 10\right)\cdot 19^{3} + \left(18 a + 12\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 a + 17 + 13 a\cdot 19 + \left(15 a + 7\right)\cdot 19^{2} + \left(15 a + 1\right)\cdot 19^{3} + \left(9 a + 3\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a + 9 + \left(4 a + 4\right)\cdot 19 + \left(4 a + 17\right)\cdot 19^{2} + \left(10 a + 4\right)\cdot 19^{3} + 3\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 18 + 11\cdot 19 + 11\cdot 19^{2} + 3\cdot 19^{3} + 4\cdot 19^{4} +O(19^{5})\)
|
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.