# Properties

 Label 2.3332.6t3.a Dimension $2$ Group $D_{6}$ Conductor $3332$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$3332$$$$\medspace = 2^{2} \cdot 7^{2} \cdot 17$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.310862272.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.3332.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $$x^{2} + 18x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$16 a + 2 + \left(12 a + 6\right)\cdot 19 + 7\cdot 19^{2} + \left(2 a + 16\right)\cdot 19^{3} + \left(15 a + 7\right)\cdot 19^{4} + \left(10 a + 3\right)\cdot 19^{5} + \left(2 a + 12\right)\cdot 19^{6} + \left(6 a + 12\right)\cdot 19^{7} +O(19^{8})$$ 16*a + 2 + (12*a + 6)*19 + 7*19^2 + (2*a + 16)*19^3 + (15*a + 7)*19^4 + (10*a + 3)*19^5 + (2*a + 12)*19^6 + (6*a + 12)*19^7+O(19^8) $r_{ 2 }$ $=$ $$3 a + 18 + \left(6 a + 2\right)\cdot 19 + \left(18 a + 14\right)\cdot 19^{2} + \left(16 a + 17\right)\cdot 19^{3} + \left(3 a + 1\right)\cdot 19^{4} + \left(8 a + 18\right)\cdot 19^{5} + \left(16 a + 3\right)\cdot 19^{6} + \left(12 a + 16\right)\cdot 19^{7} +O(19^{8})$$ 3*a + 18 + (6*a + 2)*19 + (18*a + 14)*19^2 + (16*a + 17)*19^3 + (3*a + 1)*19^4 + (8*a + 18)*19^5 + (16*a + 3)*19^6 + (12*a + 16)*19^7+O(19^8) $r_{ 3 }$ $=$ $$9 + 10\cdot 19 + 9\cdot 19^{2} + 9\cdot 19^{3} + 19^{4} + 5\cdot 19^{5} + 17\cdot 19^{6} + 4\cdot 19^{7} +O(19^{8})$$ 9 + 10*19 + 9*19^2 + 9*19^3 + 19^4 + 5*19^5 + 17*19^6 + 4*19^7+O(19^8) $r_{ 4 }$ $=$ $$3 a + 17 + \left(6 a + 12\right)\cdot 19 + \left(18 a + 11\right)\cdot 19^{2} + \left(16 a + 2\right)\cdot 19^{3} + \left(3 a + 11\right)\cdot 19^{4} + \left(8 a + 15\right)\cdot 19^{5} + \left(16 a + 6\right)\cdot 19^{6} + \left(12 a + 6\right)\cdot 19^{7} +O(19^{8})$$ 3*a + 17 + (6*a + 12)*19 + (18*a + 11)*19^2 + (16*a + 2)*19^3 + (3*a + 11)*19^4 + (8*a + 15)*19^5 + (16*a + 6)*19^6 + (12*a + 6)*19^7+O(19^8) $r_{ 5 }$ $=$ $$16 a + 1 + \left(12 a + 16\right)\cdot 19 + 4\cdot 19^{2} + \left(2 a + 1\right)\cdot 19^{3} + \left(15 a + 17\right)\cdot 19^{4} + 10 a\cdot 19^{5} + \left(2 a + 15\right)\cdot 19^{6} + \left(6 a + 2\right)\cdot 19^{7} +O(19^{8})$$ 16*a + 1 + (12*a + 16)*19 + 4*19^2 + (2*a + 1)*19^3 + (15*a + 17)*19^4 + 10*a*19^5 + (2*a + 15)*19^6 + (6*a + 2)*19^7+O(19^8) $r_{ 6 }$ $=$ $$10 + 8\cdot 19 + 9\cdot 19^{2} + 9\cdot 19^{3} + 17\cdot 19^{4} + 13\cdot 19^{5} + 19^{6} + 14\cdot 19^{7} +O(19^{8})$$ 10 + 8*19 + 9*19^2 + 9*19^3 + 17*19^4 + 13*19^5 + 19^6 + 14*19^7+O(19^8)

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

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

### Character values on conjugacy classes

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