# Properties

 Label 2.104.6t3.a Dimension 2 Group $D_{6}$ Conductor $2^{3} \cdot 13$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $104= 2^{3} \cdot 13$ Artin number field: Splitting field of $f= x^{6} + 2 x^{4} - 2 x^{3} + 2 x^{2} + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{6}$ Parity: Odd Projective image: $S_3$ Projective field: Galois closure of 3.1.104.1

## Galois action

### Roots of defining polynomial

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

### 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.