# Properties

 Label 2.475.8t8.a Dimension $2$ Group $QD_{16}$ Conductor $475$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $QD_{16}$ Conductor: $$475$$$$\medspace = 5^{2} \cdot 19$$ Artin number field: Galois closure of 8.2.107171875.1 Galois orbit size: $2$ Smallest permutation container: $QD_{16}$ 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 $\Q_{ 199 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$3 + 86\cdot 199 + 183\cdot 199^{2} + 102\cdot 199^{3} + 51\cdot 199^{4} +O(199^{5})$$ 3 + 86*199 + 183*199^2 + 102*199^3 + 51*199^4+O(199^5) $r_{ 2 }$ $=$ $$30 + 64\cdot 199 + 101\cdot 199^{2} + 44\cdot 199^{3} + 158\cdot 199^{4} +O(199^{5})$$ 30 + 64*199 + 101*199^2 + 44*199^3 + 158*199^4+O(199^5) $r_{ 3 }$ $=$ $$42 + 179\cdot 199 + 134\cdot 199^{2} + 41\cdot 199^{3} + 53\cdot 199^{4} +O(199^{5})$$ 42 + 179*199 + 134*199^2 + 41*199^3 + 53*199^4+O(199^5) $r_{ 4 }$ $=$ $$83 + 38\cdot 199 + 130\cdot 199^{2} + 97\cdot 199^{3} + 9\cdot 199^{4} +O(199^{5})$$ 83 + 38*199 + 130*199^2 + 97*199^3 + 9*199^4+O(199^5) $r_{ 5 }$ $=$ $$149 + 176\cdot 199 + 136\cdot 199^{2} + 11\cdot 199^{3} + 129\cdot 199^{4} +O(199^{5})$$ 149 + 176*199 + 136*199^2 + 11*199^3 + 129*199^4+O(199^5) $r_{ 6 }$ $=$ $$156 + 18\cdot 199 + 141\cdot 199^{2} + 103\cdot 199^{3} + 124\cdot 199^{4} +O(199^{5})$$ 156 + 18*199 + 141*199^2 + 103*199^3 + 124*199^4+O(199^5) $r_{ 7 }$ $=$ $$164 + 96\cdot 199 + 146\cdot 199^{2} + 185\cdot 199^{3} + 8\cdot 199^{4} +O(199^{5})$$ 164 + 96*199 + 146*199^2 + 185*199^3 + 8*199^4+O(199^5) $r_{ 8 }$ $=$ $$171 + 135\cdot 199 + 20\cdot 199^{2} + 9\cdot 199^{3} + 62\cdot 199^{4} +O(199^{5})$$ 171 + 135*199 + 20*199^2 + 9*199^3 + 62*199^4+O(199^5)

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

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

### Character values on conjugacy classes

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