# Properties

 Label 1.89.8t1.a.b Dimension 1 Group $C_8$ Conductor $89$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_8$ Conductor: $89$ Artin number field: Splitting field of $f= x^{8} - x^{7} + 6 x^{6} - 46 x^{5} - 143 x^{4} + 575 x^{3} + 1160 x^{2} - 16 x + 512$ over $\Q$ Size of Galois orbit: 4 Smallest containing permutation representation: $C_8$ Parity: Odd Corresponding Dirichlet character: $$\chi_{89}(37,\cdot)$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $3 + 41\cdot 97 + 84\cdot 97^{2} + 84\cdot 97^{3} + 29\cdot 97^{4} +O\left(97^{ 5 }\right)$ $r_{ 2 }$ $=$ $31 + 45\cdot 97 + 20\cdot 97^{2} + 76\cdot 97^{3} + 81\cdot 97^{4} +O\left(97^{ 5 }\right)$ $r_{ 3 }$ $=$ $60 + 17\cdot 97 + 84\cdot 97^{2} + 62\cdot 97^{3} + 4\cdot 97^{4} +O\left(97^{ 5 }\right)$ $r_{ 4 }$ $=$ $67 + 6\cdot 97 + 69\cdot 97^{2} + 12\cdot 97^{3} + 38\cdot 97^{4} +O\left(97^{ 5 }\right)$ $r_{ 5 }$ $=$ $68 + 33\cdot 97 + 44\cdot 97^{2} + 11\cdot 97^{3} + 26\cdot 97^{4} +O\left(97^{ 5 }\right)$ $r_{ 6 }$ $=$ $81 + 33\cdot 97 + 56\cdot 97^{2} + 19\cdot 97^{3} + 95\cdot 97^{4} +O\left(97^{ 5 }\right)$ $r_{ 7 }$ $=$ $83 + 8\cdot 97 + 73\cdot 97^{2} + 7\cdot 97^{3} + 37\cdot 97^{4} +O\left(97^{ 5 }\right)$ $r_{ 8 }$ $=$ $93 + 6\cdot 97 + 53\cdot 97^{2} + 15\cdot 97^{3} + 75\cdot 97^{4} +O\left(97^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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