# Properties

 Label 1.85.8t1.c.a Dimension 1 Group $C_8$ Conductor $5 \cdot 17$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_8$ Conductor: $85= 5 \cdot 17$ Artin number field: Splitting field of 8.0.6411541765625.2 defined by $f= x^{8} - x^{7} + 10 x^{6} + 6 x^{5} + 49 x^{4} - 129 x^{3} + 500 x^{2} + 2044 x + 1616$ over $\Q$ Size of Galois orbit: 4 Smallest containing permutation representation: $C_8$ Parity: Odd Corresponding Dirichlet character: $$\chi_{85}(42,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $4 + 13\cdot 43^{2} + 42\cdot 43^{3} + 39\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 2 }$ $=$ $13 + 14\cdot 43 + 22\cdot 43^{2} + 22\cdot 43^{3} + 13\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 3 }$ $=$ $23 + 29\cdot 43 + 41\cdot 43^{2} + 41\cdot 43^{3} + 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 4 }$ $=$ $24 + 32\cdot 43 + 21\cdot 43^{2} + 43^{3} + 29\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 5 }$ $=$ $36 + 27\cdot 43 + 21\cdot 43^{2} + 11\cdot 43^{3} + 16\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 6 }$ $=$ $37 + 31\cdot 43 + 3\cdot 43^{2} + 33\cdot 43^{3} + 3\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 7 }$ $=$ $39 + 40\cdot 43 + 35\cdot 43^{2} + 43^{3} + 8\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 8 }$ $=$ $40 + 37\cdot 43 + 11\cdot 43^{2} + 17\cdot 43^{3} + 16\cdot 43^{4} +O\left(43^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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