# Properties

 Label 1.2e5.8t1.2c4 Dimension 1 Group $C_8$ Conductor $2^{5}$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_8$ Conductor: $32= 2^{5}$ Artin number field: Splitting field of $f= x^{8} + 8 x^{6} + 20 x^{4} + 16 x^{2} + 2$ over $\Q$ Size of Galois orbit: 4 Smallest containing permutation representation: $C_8$ Parity: Odd Corresponding Dirichlet character: $$\chi_{32}(11,\cdot)$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $1 + 40\cdot 47 + 5\cdot 47^{2} + 20\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 2 }$ $=$ $4 + 5\cdot 47 + 41\cdot 47^{2} + 19\cdot 47^{3} + 16\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 3 }$ $=$ $11 + 13\cdot 47 + 8\cdot 47^{2} + 22\cdot 47^{3} + 20\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 4 }$ $=$ $18 + 25\cdot 47 + 45\cdot 47^{3} + 35\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 5 }$ $=$ $29 + 21\cdot 47 + 46\cdot 47^{2} + 47^{3} + 11\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 6 }$ $=$ $36 + 33\cdot 47 + 38\cdot 47^{2} + 24\cdot 47^{3} + 26\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 7 }$ $=$ $43 + 41\cdot 47 + 5\cdot 47^{2} + 27\cdot 47^{3} + 30\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 8 }$ $=$ $46 + 6\cdot 47 + 41\cdot 47^{2} + 46\cdot 47^{3} + 26\cdot 47^{4} +O\left(47^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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