# Properties

 Label 2.1080.6t3.f Dimension $2$ Group $D_{6}$ Conductor $1080$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$1080$$$$\medspace = 2^{3} \cdot 3^{3} \cdot 5$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.5832000.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.1080.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $$x^{2} + 18x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$16 + 14\cdot 19 + 5\cdot 19^{2} + 4\cdot 19^{3} + 13\cdot 19^{4} + 17\cdot 19^{5} +O(19^{6})$$ 16 + 14*19 + 5*19^2 + 4*19^3 + 13*19^4 + 17*19^5+O(19^6) $r_{ 2 }$ $=$ $$6 a + 8 + \left(7 a + 1\right)\cdot 19 + \left(9 a + 15\right)\cdot 19^{2} + 3 a\cdot 19^{3} + \left(16 a + 6\right)\cdot 19^{4} + \left(5 a + 15\right)\cdot 19^{5} +O(19^{6})$$ 6*a + 8 + (7*a + 1)*19 + (9*a + 15)*19^2 + 3*a*19^3 + (16*a + 6)*19^4 + (5*a + 15)*19^5+O(19^6) $r_{ 3 }$ $=$ $$9 + 8\cdot 19 + 3\cdot 19^{2} + 10\cdot 19^{3} + 5\cdot 19^{4} + 5\cdot 19^{5} +O(19^{6})$$ 9 + 8*19 + 3*19^2 + 10*19^3 + 5*19^4 + 5*19^5+O(19^6) $r_{ 4 }$ $=$ $$3 a + 13 + \left(8 a + 2\right)\cdot 19 + \left(10 a + 16\right)\cdot 19^{2} + \left(5 a + 6\right)\cdot 19^{3} + \left(17 a + 10\right)\cdot 19^{4} + \left(9 a + 10\right)\cdot 19^{5} +O(19^{6})$$ 3*a + 13 + (8*a + 2)*19 + (10*a + 16)*19^2 + (5*a + 6)*19^3 + (17*a + 10)*19^4 + (9*a + 10)*19^5+O(19^6) $r_{ 5 }$ $=$ $$16 a + 16 + \left(10 a + 7\right)\cdot 19 + \left(8 a + 18\right)\cdot 19^{2} + \left(13 a + 1\right)\cdot 19^{3} + \left(a + 3\right)\cdot 19^{4} + \left(9 a + 3\right)\cdot 19^{5} +O(19^{6})$$ 16*a + 16 + (10*a + 7)*19 + (8*a + 18)*19^2 + (13*a + 1)*19^3 + (a + 3)*19^4 + (9*a + 3)*19^5+O(19^6) $r_{ 6 }$ $=$ $$13 a + 14 + \left(11 a + 2\right)\cdot 19 + \left(9 a + 17\right)\cdot 19^{2} + \left(15 a + 13\right)\cdot 19^{3} + \left(2 a + 18\right)\cdot 19^{4} + \left(13 a + 4\right)\cdot 19^{5} +O(19^{6})$$ 13*a + 14 + (11*a + 2)*19 + (9*a + 17)*19^2 + (15*a + 13)*19^3 + (2*a + 18)*19^4 + (13*a + 4)*19^5+O(19^6)

### Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(2,6)(4,5)$ $(1,3)(2,5)(4,6)$ $(1,2)(3,5)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $1$ $1$ $()$ $2$ $1$ $2$ $(1,3)(2,5)(4,6)$ $-2$ $3$ $2$ $(1,2)(3,5)$ $0$ $3$ $2$ $(1,5)(2,3)(4,6)$ $0$ $2$ $3$ $(1,6,2)(3,4,5)$ $-1$ $2$ $6$ $(1,4,2,3,6,5)$ $1$
The blue line marks the conjugacy class containing complex conjugation.