# Properties

 Label 2.2004.8t6.b.a Dimension 2 Group $D_{8}$ Conductor $2^{2} \cdot 3 \cdot 167$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{8}$ Conductor: $2004= 2^{2} \cdot 3 \cdot 167$ Artin number field: Splitting field of 8.2.96577152768.2 defined by $f= x^{8} - 11 x^{6} + 64 x^{4} - 60 x^{3} - 15 x^{2} - 36 x + 9$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $D_{8}$ Parity: Odd Determinant: 1.2004.2t1.a.a Projective image: $D_4$ Projective field: Galois closure of 4.2.24048.2

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 179 }$ to precision 6.
Roots:
 $r_{ 1 }$ $=$ $6 + 112\cdot 179 + 43\cdot 179^{3} + 106\cdot 179^{4} + 17\cdot 179^{5} +O\left(179^{ 6 }\right)$ $r_{ 2 }$ $=$ $23 + 40\cdot 179 + 85\cdot 179^{2} + 30\cdot 179^{3} + 29\cdot 179^{4} + 110\cdot 179^{5} +O\left(179^{ 6 }\right)$ $r_{ 3 }$ $=$ $35 + 131\cdot 179 + 21\cdot 179^{2} + 119\cdot 179^{3} + 66\cdot 179^{4} + 122\cdot 179^{5} +O\left(179^{ 6 }\right)$ $r_{ 4 }$ $=$ $86 + 47\cdot 179 + 67\cdot 179^{2} + 26\cdot 179^{3} + 131\cdot 179^{4} + 34\cdot 179^{5} +O\left(179^{ 6 }\right)$ $r_{ 5 }$ $=$ $96 + 140\cdot 179 + 157\cdot 179^{2} + 153\cdot 179^{3} + 93\cdot 179^{4} + 140\cdot 179^{5} +O\left(179^{ 6 }\right)$ $r_{ 6 }$ $=$ $135 + 8\cdot 179 + 158\cdot 179^{2} + 164\cdot 179^{3} + 108\cdot 179^{4} + 111\cdot 179^{5} +O\left(179^{ 6 }\right)$ $r_{ 7 }$ $=$ $166 + 154\cdot 179 + 163\cdot 179^{2} + 125\cdot 179^{3} + 70\cdot 179^{4} + 116\cdot 179^{5} +O\left(179^{ 6 }\right)$ $r_{ 8 }$ $=$ $169 + 80\cdot 179 + 61\cdot 179^{2} + 52\cdot 179^{3} + 109\cdot 179^{4} + 62\cdot 179^{5} +O\left(179^{ 6 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

 Cycle notation $(1,3,2,5)(4,6,8,7)$ $(1,6,3,8,2,7,5,4)$ $(1,7)(2,6)(3,8)(4,5)$ $(1,2)(3,5)(4,8)(6,7)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,2)(3,5)(4,8)(6,7)$ $-2$ $4$ $2$ $(1,2)(4,7)(6,8)$ $0$ $4$ $2$ $(1,7)(2,6)(3,8)(4,5)$ $0$ $2$ $4$ $(1,3,2,5)(4,6,8,7)$ $0$ $2$ $8$ $(1,6,3,8,2,7,5,4)$ $-\zeta_{8}^{3} + \zeta_{8}$ $2$ $8$ $(1,8,5,6,2,4,3,7)$ $\zeta_{8}^{3} - \zeta_{8}$
The blue line marks the conjugacy class containing complex conjugation.