Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(12992961600\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5^{2} \cdot 67^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.12864000.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T34 |
| Parity: | even |
| Determinant: | 1.201.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.2.12864000.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 6x^{4} - 2x^{3} + 4x^{2} - 4x + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 6 a + 10 + 10\cdot 53 + \left(16 a + 13\right)\cdot 53^{2} + \left(19 a + 27\right)\cdot 53^{3} + \left(34 a + 44\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 45 a + 24 + \left(18 a + 41\right)\cdot 53 + \left(5 a + 32\right)\cdot 53^{2} + \left(10 a + 31\right)\cdot 53^{3} + \left(20 a + 45\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 22 + 44\cdot 53 + 40\cdot 53^{2} + 33\cdot 53^{3} + 9\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 24 + 39\cdot 53 + 12\cdot 53^{2} + 17\cdot 53^{3} + 45\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a + 45 + \left(34 a + 18\right)\cdot 53 + \left(47 a + 35\right)\cdot 53^{2} + \left(42 a + 13\right)\cdot 53^{3} + \left(32 a + 10\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 47 a + 34 + \left(52 a + 4\right)\cdot 53 + \left(36 a + 24\right)\cdot 53^{2} + \left(33 a + 35\right)\cdot 53^{3} + \left(18 a + 3\right)\cdot 53^{4} +O(53^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(3,6)$ | $-2$ |
| $9$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,3,6)$ | $1$ |
| $4$ | $3$ | $(1,3,6)(2,4,5)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,5,6,4)$ | $0$ |
| $12$ | $6$ | $(1,4,3,5,6,2)$ | $0$ |
| $12$ | $6$ | $(2,4,5)(3,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.