# Properties

 Label 2.68.8t17.a.a Dimension 2 Group $C_4\wr C_2$ Conductor $2^{2} \cdot 17$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $2$ Group: $C_4\wr C_2$ Conductor: $68= 2^{2} \cdot 17$ Artin number field: Splitting field of $f= x^{8} - 2 x^{7} + 4 x^{5} - 4 x^{4} + 3 x^{2} - 2 x + 1$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_4\wr C_2$ Parity: Odd Determinant: 1.68.4t1.a.b

## Galois action

### Roots of defining polynomial

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

### 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 value $1$ $1$ $()$ $2$ $1$ $2$ $(1,4)(2,3)(5,6)(7,8)$ $-2$ $2$ $2$ $(2,3)(7,8)$ $0$ $4$ $2$ $(1,2)(3,4)(5,7)(6,8)$ $0$ $1$ $4$ $(1,5,4,6)(2,7,3,8)$ $-2 \zeta_{4}$ $1$ $4$ $(1,6,4,5)(2,8,3,7)$ $2 \zeta_{4}$ $2$ $4$ $(2,7,3,8)$ $-\zeta_{4} + 1$ $2$ $4$ $(2,8,3,7)$ $\zeta_{4} + 1$ $2$ $4$ $(1,5,4,6)(2,3)(7,8)$ $-\zeta_{4} - 1$ $2$ $4$ $(1,6,4,5)(2,3)(7,8)$ $\zeta_{4} - 1$ $2$ $4$ $(1,5,4,6)(2,8,3,7)$ $0$ $4$ $4$ $(1,8,4,7)(2,6,3,5)$ $0$ $4$ $8$ $(1,2,6,8,4,3,5,7)$ $0$ $4$ $8$ $(1,8,5,2,4,7,6,3)$ $0$
The blue line marks the conjugacy class containing complex conjugation.