# Properties

 Label 2.2e2_29.6t3.1 Dimension 2 Group $D_{6}$ Conductor $2^{2} \cdot 29$ Frobenius-Schur indicator 1

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## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $116= 2^{2} \cdot 29$ Artin number field: Splitting field of $f= x^{6} - 2 x^{3} + x^{2} + 2 x + 2$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{6}$ Parity: Odd

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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