Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(10449\)\(\medspace = 3^{5} \cdot 43 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.31347.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.129.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.31347.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{4} - 2x^{3} + 3x^{2} + 3x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 a + 2 + \left(7 a + 11\right)\cdot 13 + \left(5 a + 4\right)\cdot 13^{2} + \left(5 a + 1\right)\cdot 13^{3} + \left(4 a + 9\right)\cdot 13^{4} +O(13^{5})\)
| $r_{ 2 }$ |
$=$ |
\( a + 7 + \left(a + 1\right)\cdot 13 + \left(7 a + 10\right)\cdot 13^{2} + \left(3 a + 11\right)\cdot 13^{3} + \left(6 a + 8\right)\cdot 13^{4} +O(13^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 11 + 9\cdot 13 + 12\cdot 13^{2} + 5\cdot 13^{3} + 5\cdot 13^{4} +O(13^{5})\)
| $r_{ 4 }$ |
$=$ |
\( a + 1 + \left(5 a + 7\right)\cdot 13 + \left(7 a + 2\right)\cdot 13^{2} + \left(7 a + 1\right)\cdot 13^{3} + \left(8 a + 8\right)\cdot 13^{4} +O(13^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 10 + 7\cdot 13 + 5\cdot 13^{2} + 10\cdot 13^{3} + 8\cdot 13^{4} +O(13^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 12 a + 8 + \left(11 a + 1\right)\cdot 13 + \left(5 a + 3\right)\cdot 13^{2} + \left(9 a + 8\right)\cdot 13^{3} + \left(6 a + 11\right)\cdot 13^{4} +O(13^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(3,6)$ | $2$ |
| $9$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,4,5)(2,3,6)$ | $-2$ |
| $4$ | $3$ | $(1,4,5)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,5,6,4)$ | $0$ |
| $12$ | $6$ | $(1,3,4,6,5,2)$ | $0$ |
| $12$ | $6$ | $(1,4,5)(3,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.