# Properties

 Label 2.165.8t17.d Dimension $2$ Group $C_4\wr C_2$ Conductor $165$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $C_4\wr C_2$ Conductor: $$165$$$$\medspace = 3 \cdot 5 \cdot 11$$ Artin number field: Galois closure of 8.0.16471125.2 Galois orbit size: $2$ Smallest permutation container: $C_4\wr C_2$ Parity: odd Projective image: $D_4$ Projective field: 4.2.12375.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 269 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$52 + 39\cdot 269 + 148\cdot 269^{2} + 164\cdot 269^{3} + 60\cdot 269^{4} +O(269^{5})$$ $r_{ 2 }$ $=$ $$60 + 178\cdot 269 + 199\cdot 269^{2} + 152\cdot 269^{3} + 180\cdot 269^{4} +O(269^{5})$$ $r_{ 3 }$ $=$ $$70 + 90\cdot 269 + 188\cdot 269^{2} + 123\cdot 269^{3} + 265\cdot 269^{4} +O(269^{5})$$ $r_{ 4 }$ $=$ $$73 + 25\cdot 269 + 98\cdot 269^{2} + 200\cdot 269^{3} + 195\cdot 269^{4} +O(269^{5})$$ $r_{ 5 }$ $=$ $$128 + 88\cdot 269 + 56\cdot 269^{2} + 41\cdot 269^{3} + 179\cdot 269^{4} +O(269^{5})$$ $r_{ 6 }$ $=$ $$130 + 246\cdot 269 + 231\cdot 269^{2} + 258\cdot 269^{3} + 263\cdot 269^{4} +O(269^{5})$$ $r_{ 7 }$ $=$ $$145 + 70\cdot 269 + 16\cdot 269^{2} + 109\cdot 269^{3} + 186\cdot 269^{4} +O(269^{5})$$ $r_{ 8 }$ $=$ $$150 + 68\cdot 269 + 137\cdot 269^{2} + 25\cdot 269^{3} + 13\cdot 269^{4} +O(269^{5})$$

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

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

### Character values on conjugacy classes

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