Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(3313408\)\(\medspace = 2^{8} \cdot 7 \cdot 43^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.471968.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | odd |
| Determinant: | 1.7.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.471968.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 2x^{4} + 2x^{2} - 4x + 4 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 a + 5 + \left(10 a + 16\right)\cdot 29 + \left(16 a + 7\right)\cdot 29^{2} + \left(12 a + 8\right)\cdot 29^{3} + \left(a + 2\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 + 8\cdot 29 + 14\cdot 29^{2} + 23\cdot 29^{3} + 18\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 2 + \left(16 a + 2\right)\cdot 29 + \left(27 a + 4\right)\cdot 29^{2} + \left(7 a + 4\right)\cdot 29^{3} + \left(5 a + 22\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 21 a + 16 + \left(18 a + 1\right)\cdot 29 + \left(12 a + 21\right)\cdot 29^{2} + \left(16 a + 25\right)\cdot 29^{3} + \left(27 a + 25\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 17 + 7\cdot 29 + 29^{2} + 9\cdot 29^{3} + 6\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 27 a + 12 + \left(12 a + 22\right)\cdot 29 + \left(a + 9\right)\cdot 29^{2} + \left(21 a + 16\right)\cdot 29^{3} + \left(23 a + 11\right)\cdot 29^{4} +O(29^{5})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(2,3)$ | $0$ |
| $9$ | $2$ | $(1,4)(2,3)$ | $0$ |
| $4$ | $3$ | $(2,3,6)$ | $-2$ |
| $4$ | $3$ | $(1,4,5)(2,3,6)$ | $1$ |
| $18$ | $4$ | $(1,2,4,3)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,2,4,3,5,6)$ | $-1$ |
| $12$ | $6$ | $(1,4,5)(2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.