Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(20725\)\(\medspace = 5^{2} \cdot 829 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.103625.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.829.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.2.103625.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + x^{4} + x^{3} + 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 29 + 11\cdot 41 + 18\cdot 41^{2} + 29\cdot 41^{3} + 28\cdot 41^{4} +O(41^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 30 + 27\cdot 41 + 2\cdot 41^{2} + 29\cdot 41^{3} + 35\cdot 41^{4} +O(41^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 18 a + 3 + \left(13 a + 35\right)\cdot 41 + \left(26 a + 8\right)\cdot 41^{2} + \left(36 a + 10\right)\cdot 41^{3} + \left(4 a + 31\right)\cdot 41^{4} +O(41^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 23 a + 16 + \left(27 a + 16\right)\cdot 41 + \left(14 a + 33\right)\cdot 41^{2} + \left(4 a + 11\right)\cdot 41^{3} + \left(36 a + 9\right)\cdot 41^{4} +O(41^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 4 a + 17 + \left(34 a + 28\right)\cdot 41 + \left(13 a + 5\right)\cdot 41^{2} + \left(9 a + 14\right)\cdot 41^{3} + \left(12 a + 36\right)\cdot 41^{4} +O(41^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 37 a + 29 + \left(6 a + 3\right)\cdot 41 + \left(27 a + 13\right)\cdot 41^{2} + \left(31 a + 28\right)\cdot 41^{3} + \left(28 a + 22\right)\cdot 41^{4} +O(41^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(3,4)$ | $2$ |
| $9$ | $2$ | $(3,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,3,4)$ | $1$ |
| $4$ | $3$ | $(1,3,4)(2,5,6)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,5,3,6,4,2)$ | $0$ |
| $12$ | $6$ | $(2,5,6)(3,4)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.