Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(2539732\)\(\medspace = 2^{2} \cdot 13^{3} \cdot 17^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.255476.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T34 |
| Parity: | even |
| Determinant: | 1.13.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.2.255476.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 2x^{4} + x^{3} + 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{2} + 42x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 a + 16 + \left(36 a + 2\right)\cdot 43 + \left(15 a + 28\right)\cdot 43^{2} + \left(37 a + 6\right)\cdot 43^{3} + \left(16 a + 27\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 28 a + 31 + \left(6 a + 23\right)\cdot 43 + \left(27 a + 7\right)\cdot 43^{2} + \left(5 a + 28\right)\cdot 43^{3} + \left(26 a + 6\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 30 + 41\cdot 43 + 28\cdot 43^{2} + 16\cdot 43^{3} + 34\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 12 + 33\cdot 43 + 13\cdot 43^{2} + 25\cdot 43^{3} + 31\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 30 a + 27 + \left(21 a + 39\right)\cdot 43 + \left(30 a + 20\right)\cdot 43^{2} + \left(28 a + 5\right)\cdot 43^{3} + \left(6 a + 4\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 13 a + 14 + \left(21 a + 31\right)\cdot 43 + \left(12 a + 29\right)\cdot 43^{2} + \left(14 a + 3\right)\cdot 43^{3} + \left(36 a + 25\right)\cdot 43^{4} +O(43^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,4)(2,5)(3,6)$ | $0$ |
| $6$ | $2$ | $(2,3)$ | $-2$ |
| $9$ | $2$ | $(2,3)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
| $4$ | $3$ | $(1,2,3)$ | $1$ |
| $18$ | $4$ | $(1,4)(2,6,3,5)$ | $0$ |
| $12$ | $6$ | $(1,5,2,6,3,4)$ | $0$ |
| $12$ | $6$ | $(2,3)(4,5,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.