Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(730644633\)\(\medspace = 3^{2} \cdot 433^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.11691.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T34 |
| Parity: | even |
| Determinant: | 1.433.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.11691.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - x^{2} + x + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 10 a + 11 + \left(4 a + 7\right)\cdot 13 + \left(4 a + 3\right)\cdot 13^{2} + \left(8 a + 4\right)\cdot 13^{3} + \left(10 a + 11\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( a + 3 + 9\cdot 13 + \left(10 a + 9\right)\cdot 13^{2} + \left(4 a + 9\right)\cdot 13^{3} + \left(7 a + 2\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 + 10\cdot 13 + 2\cdot 13^{2} + 2\cdot 13^{3} + 10\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 + 10\cdot 13^{3} + 8\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 a + 4 + \left(12 a + 8\right)\cdot 13 + \left(2 a + 6\right)\cdot 13^{2} + \left(8 a + 4\right)\cdot 13^{3} + \left(5 a + 5\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 a + 8 + \left(8 a + 2\right)\cdot 13 + \left(8 a + 3\right)\cdot 13^{2} + \left(4 a + 8\right)\cdot 13^{3} + 2 a\cdot 13^{4} +O(13^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(3,5)$ | $-2$ |
| $9$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,4,6)(2,3,5)$ | $-2$ |
| $4$ | $3$ | $(1,4,6)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,3,4,5,6,2)$ | $0$ |
| $12$ | $6$ | $(1,4,6)(3,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.