# Properties

 Label 2.2475.8t8.a Dimension $2$ Group $QD_{16}$ Conductor $2475$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $QD_{16}$ Conductor: $$2475$$$$\medspace = 3^{2} \cdot 5^{2} \cdot 11$$ Artin number field: Galois closure of 8.2.15160921875.1 Galois orbit size: $2$ Smallest permutation container: $QD_{16}$ Parity: odd Projective image: $D_4$ Projective field: 4.2.2475.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$22 + 34\cdot 89 + 41\cdot 89^{2} + 48\cdot 89^{3} + 13\cdot 89^{4} +O(89^{5})$$ $r_{ 2 }$ $=$ $$26 + 15\cdot 89 + 70\cdot 89^{2} + 32\cdot 89^{3} + 60\cdot 89^{4} +O(89^{5})$$ $r_{ 3 }$ $=$ $$38 + 55\cdot 89 + 15\cdot 89^{2} + 35\cdot 89^{3} + 70\cdot 89^{4} +O(89^{5})$$ $r_{ 4 }$ $=$ $$43 + 58\cdot 89 + 46\cdot 89^{2} + 69\cdot 89^{3} + 85\cdot 89^{4} +O(89^{5})$$ $r_{ 5 }$ $=$ $$47 + 71\cdot 89 + 65\cdot 89^{2} + 17\cdot 89^{3} + 76\cdot 89^{4} +O(89^{5})$$ $r_{ 6 }$ $=$ $$49 + 64\cdot 89 + 82\cdot 89^{2} + 65\cdot 89^{3} + 34\cdot 89^{4} +O(89^{5})$$ $r_{ 7 }$ $=$ $$55 + 41\cdot 89^{2} + 75\cdot 89^{3} + 88\cdot 89^{4} +O(89^{5})$$ $r_{ 8 }$ $=$ $$79 + 55\cdot 89 + 81\cdot 89^{2} + 10\cdot 89^{3} + 15\cdot 89^{4} +O(89^{5})$$

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

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

### Character values on conjugacy classes

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