Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(17984\)\(\medspace = 2^{6} \cdot 281 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.143872.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Projective image: | $\SOPlus(4,2)$ |
| Projective field: | Galois closure of 6.0.143872.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{2} + 16x + 3 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 14 a + 4 + \left(15 a + 4\right)\cdot 17 + \left(5 a + 1\right)\cdot 17^{2} + \left(14 a + 14\right)\cdot 17^{3} + \left(7 a + 2\right)\cdot 17^{4} +O(17^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 3 a + 1 + \left(a + 6\right)\cdot 17 + \left(11 a + 8\right)\cdot 17^{2} + \left(2 a + 5\right)\cdot 17^{3} + \left(9 a + 13\right)\cdot 17^{4} +O(17^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 11 a + \left(4 a + 10\right)\cdot 17 + \left(8 a + 15\right)\cdot 17^{2} + \left(3 a + 3\right)\cdot 17^{3} + \left(3 a + 13\right)\cdot 17^{4} +O(17^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 6 a + 11 + \left(12 a + 3\right)\cdot 17 + \left(8 a + 2\right)\cdot 17^{2} + \left(13 a + 16\right)\cdot 17^{3} + \left(13 a + 12\right)\cdot 17^{4} +O(17^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 14 + 4\cdot 17 + 11\cdot 17^{2} + 14\cdot 17^{3} + 16\cdot 17^{4} +O(17^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 6 + 5\cdot 17 + 12\cdot 17^{2} + 13\cdot 17^{3} + 8\cdot 17^{4} +O(17^{5})\)
| |
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(2,6)$ | $2$ |
| $9$ | $2$ | $(2,6)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,2,6)$ | $1$ |
| $4$ | $3$ | $(1,2,6)(3,4,5)$ | $-2$ |
| $18$ | $4$ | $(1,3)(2,5,6,4)$ | $0$ |
| $12$ | $6$ | $(1,4,2,5,6,3)$ | $0$ |
| $12$ | $6$ | $(2,6)(3,4,5)$ | $-1$ |