Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(560\)\(\medspace = 2^{4} \cdot 5 \cdot 7 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.313600.1 |
| 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 $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 15 a + 27 + \left(19 a + 6\right)\cdot 37 + 32\cdot 37^{2} + \left(13 a + 9\right)\cdot 37^{3} + \left(19 a + 18\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 26 + 18\cdot 37 + 21\cdot 37^{2} + 5\cdot 37^{3} + 26\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 21 a + 4 + \left(23 a + 11\right)\cdot 37 + \left(6 a + 21\right)\cdot 37^{2} + \left(a + 24\right)\cdot 37^{3} + \left(11 a + 22\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 29 + 31\cdot 37 + 33\cdot 37^{2} + 23\cdot 37^{3} + 6\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 22 a + 13 + \left(17 a + 32\right)\cdot 37 + \left(36 a + 14\right)\cdot 37^{2} + \left(23 a + 24\right)\cdot 37^{3} + \left(17 a + 8\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 16 a + 14 + \left(13 a + 10\right)\cdot 37 + \left(30 a + 24\right)\cdot 37^{2} + \left(35 a + 22\right)\cdot 37^{3} + \left(25 a + 28\right)\cdot 37^{4} +O(37^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,4)(3,5)$ | $-2$ |
| $3$ | $2$ | $(2,3)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,5,4)(2,6,3)$ | $-1$ |
| $2$ | $6$ | $(1,2,5,6,4,3)$ | $1$ |