Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(19989\)\(\medspace = 3^{2} \cdot 2221 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.59967.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.2221.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.59967.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - x^{4} + x^{3} + 4x^{2} - 6x + 3 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 2 a + 12 + \left(32 a + 13\right)\cdot 37 + \left(8 a + 13\right)\cdot 37^{2} + \left(28 a + 30\right)\cdot 37^{3} + \left(7 a + 36\right)\cdot 37^{4} +O(37^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 10 a + 10 + \left(30 a + 3\right)\cdot 37 + \left(14 a + 35\right)\cdot 37^{2} + \left(10 a + 4\right)\cdot 37^{3} + \left(3 a + 24\right)\cdot 37^{4} +O(37^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 4 + 9\cdot 37 + 22\cdot 37^{2} + 10\cdot 37^{3} + 22\cdot 37^{4} +O(37^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 16 + 16\cdot 37 + 33\cdot 37^{2} + 9\cdot 37^{3} + 35\cdot 37^{4} +O(37^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 27 a + 13 + \left(6 a + 3\right)\cdot 37 + \left(22 a + 27\right)\cdot 37^{2} + \left(26 a + 31\right)\cdot 37^{3} + \left(33 a + 26\right)\cdot 37^{4} +O(37^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 35 a + 20 + \left(4 a + 28\right)\cdot 37 + \left(28 a + 16\right)\cdot 37^{2} + \left(8 a + 23\right)\cdot 37^{3} + \left(29 a + 2\right)\cdot 37^{4} +O(37^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(3,5)$ | $2$ |
| $9$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,4,6)(2,3,5)$ | $-2$ |
| $4$ | $3$ | $(1,4,6)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,3,4,5,6,2)$ | $0$ |
| $12$ | $6$ | $(1,4,6)(3,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.