Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(20624\)\(\medspace = 2^{4} \cdot 1289 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.82496.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.1289.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.82496.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{4} - 2x^{3} + 2x^{2} + 2x + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 2 + 15\cdot 29 + 16\cdot 29^{2} + 3\cdot 29^{3} + 16\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 + 14\cdot 29 + 26\cdot 29^{2} + 23\cdot 29^{3} + 24\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 16 a + 17 + \left(22 a + 16\right)\cdot 29 + \left(8 a + 24\right)\cdot 29^{2} + \left(16 a + 19\right)\cdot 29^{3} + \left(a + 10\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 13 a + 10 + \left(6 a + 26\right)\cdot 29 + \left(20 a + 16\right)\cdot 29^{2} + \left(12 a + 5\right)\cdot 29^{3} + \left(27 a + 2\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 26 a + 13 + \left(9 a + 24\right)\cdot 29 + \left(5 a + 21\right)\cdot 29^{2} + \left(22 a + 7\right)\cdot 29^{3} + \left(11 a + 27\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 a + 27 + \left(19 a + 18\right)\cdot 29 + \left(23 a + 9\right)\cdot 29^{2} + \left(6 a + 26\right)\cdot 29^{3} + \left(17 a + 5\right)\cdot 29^{4} +O(29^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(3,4)$ | $2$ |
| $9$ | $2$ | $(3,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,3,4)$ | $1$ |
| $4$ | $3$ | $(1,3,4)(2,5,6)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,5,3,6,4,2)$ | $0$ |
| $12$ | $6$ | $(2,5,6)(3,4)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.