# Properties

 Label 2.87.6t3.b Dimension $2$ Group $D_{6}$ Conductor $87$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$87$$$$\medspace = 3 \cdot 29$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.0.22707.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.87.1

## 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:
 $r_{ 1 }$ $=$ $$6 a + 8 + \left(7 a + 12\right)\cdot 19 + \left(15 a + 9\right)\cdot 19^{2} + \left(3 a + 10\right)\cdot 19^{3} + \left(9 a + 18\right)\cdot 19^{4} +O(19^{5})$$ 6*a + 8 + (7*a + 12)*19 + (15*a + 9)*19^2 + (3*a + 10)*19^3 + (9*a + 18)*19^4+O(19^5) $r_{ 2 }$ $=$ $$11 a + 1 + \left(2 a + 8\right)\cdot 19 + 6 a\cdot 19^{2} + \left(16 a + 17\right)\cdot 19^{3} + \left(16 a + 13\right)\cdot 19^{4} +O(19^{5})$$ 11*a + 1 + (2*a + 8)*19 + 6*a*19^2 + (16*a + 17)*19^3 + (16*a + 13)*19^4+O(19^5) $r_{ 3 }$ $=$ $$18 + 14\cdot 19 + 6\cdot 19^{2} + 12\cdot 19^{3} + 8\cdot 19^{4} +O(19^{5})$$ 18 + 14*19 + 6*19^2 + 12*19^3 + 8*19^4+O(19^5) $r_{ 4 }$ $=$ $$5 + 8\cdot 19 + 18\cdot 19^{2} + 9\cdot 19^{3} + 15\cdot 19^{4} +O(19^{5})$$ 5 + 8*19 + 18*19^2 + 9*19^3 + 15*19^4+O(19^5) $r_{ 5 }$ $=$ $$13 a + 14 + \left(11 a + 13\right)\cdot 19 + \left(3 a + 17\right)\cdot 19^{2} + \left(15 a + 17\right)\cdot 19^{3} + \left(9 a + 4\right)\cdot 19^{4} +O(19^{5})$$ 13*a + 14 + (11*a + 13)*19 + (3*a + 17)*19^2 + (15*a + 17)*19^3 + (9*a + 4)*19^4+O(19^5) $r_{ 6 }$ $=$ $$8 a + 12 + \left(16 a + 18\right)\cdot 19 + \left(12 a + 3\right)\cdot 19^{2} + \left(2 a + 8\right)\cdot 19^{3} + \left(2 a + 14\right)\cdot 19^{4} +O(19^{5})$$ 8*a + 12 + (16*a + 18)*19 + (12*a + 3)*19^2 + (2*a + 8)*19^3 + (2*a + 14)*19^4+O(19^5)

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

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

### Character values on conjugacy classes

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