Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(24128\)\(\medspace = 2^{6} \cdot 13 \cdot 29 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.96512.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.377.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.96512.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + x^{4} - 2x^{3} - x^{2} + 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 + 13\cdot 53^{2} + 25\cdot 53^{3} + 13\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 28 a + 4 + \left(35 a + 44\right)\cdot 53 + \left(40 a + 40\right)\cdot 53^{2} + \left(44 a + 52\right)\cdot 53^{3} + \left(11 a + 16\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 a + 4 + \left(33 a + 48\right)\cdot 53 + \left(21 a + 2\right)\cdot 53^{2} + \left(14 a + 18\right)\cdot 53^{3} + \left(37 a + 38\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 25 a + 10 + \left(17 a + 52\right)\cdot 53 + \left(12 a + 8\right)\cdot 53^{2} + \left(8 a + 32\right)\cdot 53^{3} + \left(41 a + 19\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 40 + 52\cdot 53 + 36\cdot 53^{2} + 29\cdot 53^{3} + 3\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 46 a + 32 + \left(19 a + 14\right)\cdot 53 + \left(31 a + 3\right)\cdot 53^{2} + \left(38 a + 1\right)\cdot 53^{3} + \left(15 a + 14\right)\cdot 53^{4} +O(53^{5})\)
|
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.