# Properties

 Label 2.1520.8t6.b Dimension $2$ Group $D_{8}$ Conductor $1520$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{8}$ Conductor: $$1520$$$$\medspace = 2^{4} \cdot 5 \cdot 19$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 8.2.1097440000.5 Galois orbit size: $2$ Smallest permutation container: $D_{8}$ Parity: odd Projective image: $D_4$ Projective field: Galois closure of 4.2.475.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 191 }$ to precision 6.
Roots:
 $r_{ 1 }$ $=$ $$13 + 22\cdot 191 + 117\cdot 191^{2} + 125\cdot 191^{3} + 136\cdot 191^{4} + 83\cdot 191^{5} +O(191^{6})$$ 13 + 22*191 + 117*191^2 + 125*191^3 + 136*191^4 + 83*191^5+O(191^6) $r_{ 2 }$ $=$ $$30 + 40\cdot 191 + 171\cdot 191^{2} + 106\cdot 191^{3} + 116\cdot 191^{4} + 158\cdot 191^{5} +O(191^{6})$$ 30 + 40*191 + 171*191^2 + 106*191^3 + 116*191^4 + 158*191^5+O(191^6) $r_{ 3 }$ $=$ $$37 + 26\cdot 191 + 11\cdot 191^{2} + 96\cdot 191^{3} + 144\cdot 191^{4} + 104\cdot 191^{5} +O(191^{6})$$ 37 + 26*191 + 11*191^2 + 96*191^3 + 144*191^4 + 104*191^5+O(191^6) $r_{ 4 }$ $=$ $$68 + 114\cdot 191 + 45\cdot 191^{2} + 159\cdot 191^{3} + 13\cdot 191^{4} + 77\cdot 191^{5} +O(191^{6})$$ 68 + 114*191 + 45*191^2 + 159*191^3 + 13*191^4 + 77*191^5+O(191^6) $r_{ 5 }$ $=$ $$123 + 76\cdot 191 + 145\cdot 191^{2} + 31\cdot 191^{3} + 177\cdot 191^{4} + 113\cdot 191^{5} +O(191^{6})$$ 123 + 76*191 + 145*191^2 + 31*191^3 + 177*191^4 + 113*191^5+O(191^6) $r_{ 6 }$ $=$ $$154 + 164\cdot 191 + 179\cdot 191^{2} + 94\cdot 191^{3} + 46\cdot 191^{4} + 86\cdot 191^{5} +O(191^{6})$$ 154 + 164*191 + 179*191^2 + 94*191^3 + 46*191^4 + 86*191^5+O(191^6) $r_{ 7 }$ $=$ $$161 + 150\cdot 191 + 19\cdot 191^{2} + 84\cdot 191^{3} + 74\cdot 191^{4} + 32\cdot 191^{5} +O(191^{6})$$ 161 + 150*191 + 19*191^2 + 84*191^3 + 74*191^4 + 32*191^5+O(191^6) $r_{ 8 }$ $=$ $$178 + 168\cdot 191 + 73\cdot 191^{2} + 65\cdot 191^{3} + 54\cdot 191^{4} + 107\cdot 191^{5} +O(191^{6})$$ 178 + 168*191 + 73*191^2 + 65*191^3 + 54*191^4 + 107*191^5+O(191^6)

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $2$ $2$ $1$ $2$ $(1,8)(2,7)(3,6)(4,5)$ $-2$ $-2$ $4$ $2$ $(1,7)(2,8)(4,5)$ $0$ $0$ $4$ $2$ $(1,4)(2,3)(5,8)(6,7)$ $0$ $0$ $2$ $4$ $(1,7,8,2)(3,5,6,4)$ $0$ $0$ $2$ $8$ $(1,6,7,4,8,3,2,5)$ $-\zeta_{8}^{3} + \zeta_{8}$ $\zeta_{8}^{3} - \zeta_{8}$ $2$ $8$ $(1,4,2,6,8,5,7,3)$ $\zeta_{8}^{3} - \zeta_{8}$ $-\zeta_{8}^{3} + \zeta_{8}$
The blue line marks the conjugacy class containing complex conjugation.