Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(3311\)\(\medspace = 7 \cdot 11 \cdot 43 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.76739047.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.3311.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 9\cdot 37 + 32\cdot 37^{2} + 3\cdot 37^{3} + 3\cdot 37^{4} + 8\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 26 a + 14 + \left(15 a + 19\right)\cdot 37 + \left(34 a + 1\right)\cdot 37^{2} + \left(30 a + 5\right)\cdot 37^{3} + \left(15 a + 23\right)\cdot 37^{4} + \left(36 a + 2\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 29 a + 4 + \left(4 a + 35\right)\cdot 37 + \left(4 a + 24\right)\cdot 37^{2} + \left(34 a + 4\right)\cdot 37^{3} + \left(27 a + 6\right)\cdot 37^{4} + \left(29 a + 11\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 33 + 2\cdot 37 + 3\cdot 37^{2} + 14\cdot 37^{3} + 13\cdot 37^{4} + 2\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a + 9 + \left(32 a + 25\right)\cdot 37 + \left(32 a + 36\right)\cdot 37^{2} + \left(2 a + 25\right)\cdot 37^{3} + \left(9 a + 9\right)\cdot 37^{4} + \left(7 a + 28\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 11 a + 7 + \left(21 a + 19\right)\cdot 37 + \left(2 a + 12\right)\cdot 37^{2} + \left(6 a + 20\right)\cdot 37^{3} + \left(21 a + 18\right)\cdot 37^{4} + 21\cdot 37^{5} +O(37^{6})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,4)(2,3)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,2,6)(3,4,5)$ | $-1$ |
| $2$ | $6$ | $(1,3,2,4,6,5)$ | $1$ |