Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(428288\)\(\medspace = 2^{8} \cdot 7 \cdot 239 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.3426304.3 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.1673.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.2.3426304.3 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + x^{4} - 4x^{3} + 9x^{2} - 10x + 7 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 23 a + 38 + \left(19 a + 16\right)\cdot 41 + \left(40 a + 3\right)\cdot 41^{2} + \left(27 a + 40\right)\cdot 41^{3} + \left(7 a + 28\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 a + 25 + \left(21 a + 11\right)\cdot 41 + 23\cdot 41^{2} + \left(13 a + 1\right)\cdot 41^{3} + \left(33 a + 24\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 34 + 35\cdot 41 + 37\cdot 41^{2} + 7\cdot 41^{3} + 28\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 a + 18 + \left(33 a + 5\right)\cdot 41 + \left(30 a + 7\right)\cdot 41^{2} + \left(21 a + 20\right)\cdot 41^{3} + \left(18 a + 14\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 15 a + 14 + \left(7 a + 39\right)\cdot 41 + \left(10 a + 24\right)\cdot 41^{2} + \left(19 a + 13\right)\cdot 41^{3} + \left(22 a + 7\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 37 + 13\cdot 41 + 26\cdot 41^{2} + 39\cdot 41^{3} + 19\cdot 41^{4} +O(41^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(3,4)$ | $2$ |
| $9$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $4$ | $3$ | $(1,2,6)(3,4,5)$ | $-2$ |
| $4$ | $3$ | $(1,2,6)$ | $1$ |
| $18$ | $4$ | $(1,3,2,4)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,4,2,5,6,3)$ | $0$ |
| $12$ | $6$ | $(1,2,6)(3,4)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.