# Properties

 Label 2.3e2_5_19.8t6.2c1 Dimension 2 Group $D_{8}$ Conductor $3^{2} \cdot 5 \cdot 19$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{8}$ Conductor: $855= 3^{2} \cdot 5 \cdot 19$ Artin number field: Splitting field of $f= x^{8} - x^{7} + 3 x^{5} + 9 x^{4} + 2 x^{3} + 8 x^{2} - 4 x + 7$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $D_{8}$ Parity: Odd Determinant: 1.5_19.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 191 }$ to precision 5.
Roots: \begin{aligned} r_{ 1 } &= 39 + 111\cdot 191 + 175\cdot 191^{2} + 190\cdot 191^{3} + 41\cdot 191^{4} +O\left(191^{ 5 }\right) \\ r_{ 2 } &= 54 + 71\cdot 191 + 122\cdot 191^{2} + 175\cdot 191^{3} + 108\cdot 191^{4} +O\left(191^{ 5 }\right) \\ r_{ 3 } &= 100 + 63\cdot 191 + 99\cdot 191^{2} + 63\cdot 191^{3} + 190\cdot 191^{4} +O\left(191^{ 5 }\right) \\ r_{ 4 } &= 114 + 149\cdot 191 + 64\cdot 191^{2} + 116\cdot 191^{4} +O\left(191^{ 5 }\right) \\ r_{ 5 } &= 139 + 2\cdot 191 + 128\cdot 191^{2} + 60\cdot 191^{3} + 54\cdot 191^{4} +O\left(191^{ 5 }\right) \\ r_{ 6 } &= 160 + 97\cdot 191 + 7\cdot 191^{2} + 97\cdot 191^{3} + 135\cdot 191^{4} +O\left(191^{ 5 }\right) \\ r_{ 7 } &= 163 + 9\cdot 191 + 105\cdot 191^{2} + 4\cdot 191^{3} + 101\cdot 191^{4} +O\left(191^{ 5 }\right) \\ r_{ 8 } &= 187 + 66\cdot 191 + 61\cdot 191^{2} + 171\cdot 191^{3} + 15\cdot 191^{4} +O\left(191^{ 5 }\right) \\ \end{aligned}

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

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

### Character values on conjugacy classes

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