# Properties

 Label 1.17.8t1.a.a Dimension 1 Group $C_8$ Conductor $17$ Root number not computed Frobenius-Schur indicator 0

# Learn more about

## Basic invariants

 Dimension: $1$ Group: $C_8$ Conductor: $17$ Artin number field: Splitting field of $$\Q(\zeta_{17})^+$$ defined by $f= x^{8} - x^{7} - 7 x^{6} + 6 x^{5} + 15 x^{4} - 10 x^{3} - 10 x^{2} + 4 x + 1$ over $\Q$ Size of Galois orbit: 4 Smallest containing permutation representation: $C_8$ Parity: Even Corresponding Dirichlet character: $$\chi_{17}(8,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $4 + 49\cdot 67 + 2\cdot 67^{2} + 56\cdot 67^{3} + 16\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 2 }$ $=$ $7 + 9\cdot 67 + 65\cdot 67^{2} + 22\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 3 }$ $=$ $20 + 7\cdot 67 + 12\cdot 67^{2} + 21\cdot 67^{3} + 5\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 4 }$ $=$ $34 + 55\cdot 67 + 10\cdot 67^{2} + 42\cdot 67^{3} + 58\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 5 }$ $=$ $45 + 47\cdot 67 + 37\cdot 67^{2} + 11\cdot 67^{3} + 50\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 6 }$ $=$ $52 + 61\cdot 67 + 57\cdot 67^{2} + 7\cdot 67^{3} + 53\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 7 }$ $=$ $53 + 9\cdot 67 + 56\cdot 67^{2} + 56\cdot 67^{3} + 64\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 8 }$ $=$ $54 + 27\cdot 67 + 25\cdot 67^{2} + 4\cdot 67^{3} + 64\cdot 67^{4} +O\left(67^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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