Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(1014300\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 23 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.7100100.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.92.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.7100100.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 7x^{4} + 4x^{3} + 10x^{2} + 11x + 4 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 21 a + 15 + \left(38 a + 32\right)\cdot 43 + \left(32 a + 17\right)\cdot 43^{2} + \left(14 a + 42\right)\cdot 43^{3} + \left(37 a + 41\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 31 a + 10 + \left(34 a + 21\right)\cdot 43 + \left(36 a + 21\right)\cdot 43^{2} + \left(34 a + 42\right)\cdot 43^{3} + \left(33 a + 41\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 22 a + 36 + \left(4 a + 6\right)\cdot 43 + \left(10 a + 12\right)\cdot 43^{2} + \left(28 a + 24\right)\cdot 43^{3} + \left(5 a + 21\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 + 27\cdot 43 + 9\cdot 43^{2} + 25\cdot 43^{3} + 34\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 a + 41 + \left(8 a + 24\right)\cdot 43 + \left(6 a + 23\right)\cdot 43^{2} + \left(8 a + 40\right)\cdot 43^{3} + \left(9 a + 40\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 17 + 16\cdot 43 + 43^{2} + 40\cdot 43^{3} + 33\cdot 43^{4} +O(43^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(1,3)$ | $2$ |
| $9$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $4$ | $3$ | $(1,3,6)(2,4,5)$ | $-2$ |
| $4$ | $3$ | $(2,4,5)$ | $1$ |
| $18$ | $4$ | $(1,4,3,2)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,2,3,4,6,5)$ | $0$ |
| $12$ | $6$ | $(1,3)(2,4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.