Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(2224\)\(\medspace = 2^{4} \cdot 139 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.1236544.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.139.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{2} + 16x + 3 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 9\cdot 17 + 7\cdot 17^{2} + 7\cdot 17^{3} + 5\cdot 17^{4} + 7\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 a + 5 + \left(a + 3\right)\cdot 17 + 2 a\cdot 17^{2} + \left(11 a + 2\right)\cdot 17^{3} + \left(3 a + 4\right)\cdot 17^{4} + \left(13 a + 7\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 a + 15 + \left(a + 8\right)\cdot 17 + \left(2 a + 16\right)\cdot 17^{2} + \left(11 a + 5\right)\cdot 17^{3} + \left(3 a + 3\right)\cdot 17^{4} + 13 a\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 14 + 7\cdot 17 + 9\cdot 17^{2} + 9\cdot 17^{3} + 11\cdot 17^{4} + 9\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 a + 12 + \left(15 a + 13\right)\cdot 17 + \left(14 a + 16\right)\cdot 17^{2} + \left(5 a + 14\right)\cdot 17^{3} + \left(13 a + 12\right)\cdot 17^{4} + \left(3 a + 9\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 a + 2 + \left(15 a + 8\right)\cdot 17 + 14 a\cdot 17^{2} + \left(5 a + 11\right)\cdot 17^{3} + \left(13 a + 13\right)\cdot 17^{4} + \left(3 a + 16\right)\cdot 17^{5} +O(17^{6})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,3)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,5,3)(2,6,4)$ | $-1$ |
| $2$ | $6$ | $(1,6,5,4,3,2)$ | $1$ |