# Properties

 Label 2.1328.6t3.b.a Dimension 2 Group $D_{6}$ Conductor $2^{4} \cdot 83$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $1328= 2^{4} \cdot 83$ Artin number field: Splitting field of 6.0.440896.1 defined by $f= x^{6} - x^{4} - 3 x^{2} + 4$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{6}$ Parity: Odd Determinant: 1.83.2t1.a.a Projective image: $S_3$ Projective field: Galois closure of 3.1.83.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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