# Properties

 Label 2.111.8t6.a.a Dimension 2 Group $D_{8}$ Conductor $3 \cdot 37$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{8}$ Conductor: $111= 3 \cdot 37$ Artin number field: Splitting field of 8.0.4102893.1 defined by $f= x^{8} - 3 x^{7} + 3 x^{6} - 3 x^{5} + 5 x^{4} - 6 x^{3} + 6 x^{2} - 3 x + 1$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $D_{8}$ Parity: Odd Determinant: 1.111.2t1.a.a Projective image: $D_4$ Projective field: Galois closure of 4.0.333.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $32 + 15\cdot 127 + 60\cdot 127^{2} + 78\cdot 127^{3} + 55\cdot 127^{4} +O\left(127^{ 5 }\right)$ $r_{ 2 }$ $=$ $51 + 34\cdot 127 + 51\cdot 127^{2} + 96\cdot 127^{3} + 108\cdot 127^{4} +O\left(127^{ 5 }\right)$ $r_{ 3 }$ $=$ $64 + 34\cdot 127 + 104\cdot 127^{2} + 8\cdot 127^{3} + 65\cdot 127^{4} +O\left(127^{ 5 }\right)$ $r_{ 4 }$ $=$ $89 + 3\cdot 127 + 48\cdot 127^{2} + 60\cdot 127^{3} + 53\cdot 127^{4} +O\left(127^{ 5 }\right)$ $r_{ 5 }$ $=$ $93 + 72\cdot 127 + 58\cdot 127^{2} + 57\cdot 127^{3} + 105\cdot 127^{4} +O\left(127^{ 5 }\right)$ $r_{ 6 }$ $=$ $94 + 22\cdot 127 + 57\cdot 127^{2} + 23\cdot 127^{3} + 60\cdot 127^{4} +O\left(127^{ 5 }\right)$ $r_{ 7 }$ $=$ $104 + 113\cdot 127 + 85\cdot 127^{2} + 100\cdot 127^{3} + 109\cdot 127^{4} +O\left(127^{ 5 }\right)$ $r_{ 8 }$ $=$ $111 + 83\cdot 127 + 42\cdot 127^{2} + 82\cdot 127^{3} + 76\cdot 127^{4} +O\left(127^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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