# Properties

 Label 2.68.8t17.a Dimension $2$ Group $C_4\wr C_2$ Conductor $68$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $C_4\wr C_2$ Conductor: $$68$$$$\medspace = 2^{2} \cdot 17$$ Artin number field: Galois closure of 8.0.1257728.1 Galois orbit size: $2$ Smallest permutation container: $C_4\wr C_2$ Parity: odd Projective image: $D_4$ Projective field: 4.2.19652.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$23 + 19\cdot 149 + 46\cdot 149^{2} + 81\cdot 149^{3} + 103\cdot 149^{4} +O(149^{5})$$ $r_{ 2 }$ $=$ $$61 + 34\cdot 149 + 129\cdot 149^{2} + 79\cdot 149^{3} + 110\cdot 149^{4} +O(149^{5})$$ $r_{ 3 }$ $=$ $$81 + 8\cdot 149 + 91\cdot 149^{2} + 83\cdot 149^{3} + 71\cdot 149^{4} +O(149^{5})$$ $r_{ 4 }$ $=$ $$98 + 18\cdot 149 + 67\cdot 149^{3} + 149^{4} +O(149^{5})$$ $r_{ 5 }$ $=$ $$100 + 63\cdot 149 + 100\cdot 149^{2} + 70\cdot 149^{3} + 78\cdot 149^{4} +O(149^{5})$$ $r_{ 6 }$ $=$ $$122 + 35\cdot 149 + 36\cdot 149^{2} + 22\cdot 149^{3} + 101\cdot 149^{4} +O(149^{5})$$ $r_{ 7 }$ $=$ $$124 + 114\cdot 149 + 49\cdot 149^{2} + 142\cdot 149^{3} + 36\cdot 149^{4} +O(149^{5})$$ $r_{ 8 }$ $=$ $$138 + 2\cdot 149 + 143\cdot 149^{2} + 48\cdot 149^{3} + 92\cdot 149^{4} +O(149^{5})$$

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

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

### Character values on conjugacy classes

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