Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(1815\)\(\medspace = 3 \cdot 5 \cdot 11^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.36236475.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.1815.1 |
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 $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
![Copy content]()
![Toggle raw display]()
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 a + 18 + \left(15 a + 26\right)\cdot 37 + \left(7 a + 9\right)\cdot 37^{2} + \left(12 a + 25\right)\cdot 37^{3} + a\cdot 37^{4} + \left(25 a + 11\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 32 + 17\cdot 37 + 28\cdot 37^{2} + 2\cdot 37^{3} + 27\cdot 37^{4} + 37^{5} +O(37^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 21 a + 23 + \left(31 a + 3\right)\cdot 37 + \left(2 a + 24\right)\cdot 37^{2} + \left(4 a + 14\right)\cdot 37^{3} + \left(16 a + 22\right)\cdot 37^{4} + \left(9 a + 30\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 + 27\cdot 37 + 20\cdot 37^{2} + 10\cdot 37^{3} + 21\cdot 37^{4} + 16\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 24 a + 33 + 21 a\cdot 37 + \left(29 a + 24\right)\cdot 37^{2} + \left(24 a + 29\right)\cdot 37^{3} + \left(35 a + 30\right)\cdot 37^{4} + \left(11 a + 35\right)\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 16 a + 33 + \left(5 a + 34\right)\cdot 37 + \left(34 a + 3\right)\cdot 37^{2} + \left(32 a + 28\right)\cdot 37^{3} + \left(20 a + 8\right)\cdot 37^{4} + \left(27 a + 15\right)\cdot 37^{5} +O(37^{6})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,4)(3,5)$ | $-2$ |
| $3$ | $2$ | $(2,3)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,5,4)(2,6,3)$ | $-1$ |
| $2$ | $6$ | $(1,2,5,6,4,3)$ | $1$ |