Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(8001\)\(\medspace = 3^{2} \cdot 7 \cdot 127 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.24003.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.889.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.24003.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 3x^{3} - x^{2} - 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 a + 29 + \left(20 a + 10\right)\cdot 37 + \left(21 a + 32\right)\cdot 37^{2} + \left(21 a + 9\right)\cdot 37^{3} + \left(14 a + 26\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 31 a + 1 + \left(9 a + 6\right)\cdot 37 + \left(33 a + 13\right)\cdot 37^{2} + \left(20 a + 14\right)\cdot 37^{3} + \left(3 a + 22\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 31 + 27\cdot 37 + 27\cdot 37^{2} + 36\cdot 37^{3} + 20\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 34 a + 4 + \left(16 a + 14\right)\cdot 37 + \left(15 a + 24\right)\cdot 37^{2} + 15 a\cdot 37^{3} + \left(22 a + 26\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 6 a + 14 + \left(27 a + 14\right)\cdot 37 + \left(3 a + 25\right)\cdot 37^{2} + \left(16 a + 27\right)\cdot 37^{3} + \left(33 a + 15\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 33 + 25\cdot 37^{2} + 21\cdot 37^{3} + 36\cdot 37^{4} +O(37^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(3,4)$ | $2$ |
| $9$ | $2$ | $(3,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,3,4)(2,5,6)$ | $-2$ |
| $4$ | $3$ | $(1,3,4)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,5,3,6,4,2)$ | $0$ |
| $12$ | $6$ | $(2,5,6)(3,4)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.