# Properties

 Label 2.140.6t3.b Dimension $2$ Group $D_{6}$ Conductor $140$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$140$$$$\medspace = 2^{2} \cdot 5 \cdot 7$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.98000.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.140.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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