# Properties

 Label 2.1083.6t3.a Dimension $2$ Group $D_{6}$ Conductor $1083$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$1083$$$$\medspace = 3 \cdot 19^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.0.22284891.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.1083.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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