# Properties

 Label 2.117.8t17.b.b Dimension 2 Group $C_4\wr C_2$ Conductor $3^{2} \cdot 13$ Root number not computed Frobenius-Schur indicator 0

# Learn more about

## Basic invariants

 Dimension: $2$ Group: $C_4\wr C_2$ Conductor: $117= 3^{2} \cdot 13$ Artin number field: Splitting field of $f= x^{8} - 2 x^{6} - 3 x^{5} + 3 x^{4} + 3 x^{3} - 2 x^{2} + 1$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_4\wr C_2$ Parity: Odd Determinant: 1.13.4t1.a.b

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $48 + 52\cdot 139 + 37\cdot 139^{2} + 86\cdot 139^{3} + 30\cdot 139^{4} +O\left(139^{ 5 }\right)$ $r_{ 2 }$ $=$ $55 + 24\cdot 139 + 70\cdot 139^{2} + 32\cdot 139^{3} + 93\cdot 139^{4} +O\left(139^{ 5 }\right)$ $r_{ 3 }$ $=$ $74 + 93\cdot 139 + 61\cdot 139^{2} + 116\cdot 139^{3} + 51\cdot 139^{4} +O\left(139^{ 5 }\right)$ $r_{ 4 }$ $=$ $77 + 83\cdot 139 + 15\cdot 139^{2} + 95\cdot 139^{3} + 98\cdot 139^{4} +O\left(139^{ 5 }\right)$ $r_{ 5 }$ $=$ $90 + 116\cdot 139 + 20\cdot 139^{2} + 25\cdot 139^{3} + 118\cdot 139^{4} +O\left(139^{ 5 }\right)$ $r_{ 6 }$ $=$ $93 + 94\cdot 139 + 130\cdot 139^{2} + 28\cdot 139^{3} + 62\cdot 139^{4} +O\left(139^{ 5 }\right)$ $r_{ 7 }$ $=$ $122 + 72\cdot 139 + 124\cdot 139^{2} + 31\cdot 139^{3} + 123\cdot 139^{4} +O\left(139^{ 5 }\right)$ $r_{ 8 }$ $=$ $136 + 17\cdot 139 + 95\cdot 139^{2} + 117\cdot 139^{4} +O\left(139^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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