Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(1260\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.529200.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.140.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 16 + 8\cdot 19 + 17\cdot 19^{2} + 2\cdot 19^{3} + 4\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a + 5 + \left(12 a + 14\right)\cdot 19 + \left(15 a + 8\right)\cdot 19^{2} + \left(5 a + 3\right)\cdot 19^{3} + \left(3 a + 18\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 a + 17 + \left(6 a + 14\right)\cdot 19 + \left(3 a + 11\right)\cdot 19^{2} + \left(13 a + 12\right)\cdot 19^{3} + \left(15 a + 15\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 17 + 16\cdot 19 + 5\cdot 19^{2} + 2\cdot 19^{3} + 2\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a + 16 + \left(2 a + 3\right)\cdot 19 + \left(3 a + 6\right)\cdot 19^{2} + \left(3 a + 8\right)\cdot 19^{3} + \left(4 a + 17\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 11 a + 5 + \left(16 a + 17\right)\cdot 19 + \left(15 a + 6\right)\cdot 19^{2} + \left(15 a + 8\right)\cdot 19^{3} + \left(14 a + 18\right)\cdot 19^{4} +O(19^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,5)(2,4)(3,6)$ | $0$ |
| $2$ | $3$ | $(1,3,2)(4,6,5)$ | $-1$ |
| $2$ | $6$ | $(1,6,2,4,3,5)$ | $1$ |