# Properties

 Label 2.112.8t17.a.b Dimension 2 Group $C_4\wr C_2$ Conductor $2^{4} \cdot 7$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $2$ Group: $C_4\wr C_2$ Conductor: $112= 2^{4} \cdot 7$ Artin number field: Splitting field of $f= x^{8} - 3 x^{7} + 6 x^{6} - 8 x^{5} + 10 x^{4} - 9 x^{3} + 6 x^{2} - 2 x + 1$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_4\wr C_2$ Parity: Odd Determinant: 1.112.4t1.b.b

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 239 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $30 + 93\cdot 239 + 155\cdot 239^{2} + 152\cdot 239^{3} + 154\cdot 239^{4} +O\left(239^{ 5 }\right)$ $r_{ 2 }$ $=$ $92 + 154\cdot 239 + 26\cdot 239^{2} + 233\cdot 239^{3} + 21\cdot 239^{4} +O\left(239^{ 5 }\right)$ $r_{ 3 }$ $=$ $122 + 203\cdot 239 + 211\cdot 239^{2} + 50\cdot 239^{3} + 166\cdot 239^{4} +O\left(239^{ 5 }\right)$ $r_{ 4 }$ $=$ $126 + 68\cdot 239 + 25\cdot 239^{2} + 86\cdot 239^{3} + 60\cdot 239^{4} +O\left(239^{ 5 }\right)$ $r_{ 5 }$ $=$ $187 + 73\cdot 239 + 178\cdot 239^{2} + 232\cdot 239^{3} + 208\cdot 239^{4} +O\left(239^{ 5 }\right)$ $r_{ 6 }$ $=$ $194 + 202\cdot 239 + 87\cdot 239^{2} + 65\cdot 239^{3} + 69\cdot 239^{4} +O\left(239^{ 5 }\right)$ $r_{ 7 }$ $=$ $211 + 85\cdot 239 + 207\cdot 239^{2} + 116\cdot 239^{3} + 137\cdot 239^{4} +O\left(239^{ 5 }\right)$ $r_{ 8 }$ $=$ $236 + 73\cdot 239 + 63\cdot 239^{2} + 18\cdot 239^{3} + 137\cdot 239^{4} +O\left(239^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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