# Properties

 Label 2.87.6t3.b.a Dimension 2 Group $D_{6}$ Conductor $3 \cdot 29$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $87= 3 \cdot 29$ Artin number field: Splitting field of $f= x^{6} - x^{5} + 4 x^{4} - 4 x^{3} + 5 x^{2} - 3 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{6}$ Parity: Odd Determinant: 1.87.2t1.a.a

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

### 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 value $1$ $1$ $()$ $2$ $1$ $2$ $(1,2)(3,4)(5,6)$ $-2$ $3$ $2$ $(3,6)(4,5)$ $0$ $3$ $2$ $(1,2)(3,5)(4,6)$ $0$ $2$ $3$ $(1,5,4)(2,6,3)$ $-1$ $2$ $6$ $(1,3,5,2,4,6)$ $1$
The blue line marks the conjugacy class containing complex conjugation.