# Properties

 Label 1.160.8t1.b.b Dimension $1$ Group $C_8$ Conductor $160$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_8$ Conductor: $$160$$$$\medspace = 2^{5} \cdot 5$$ Artin field: 8.0.33554432000000.1 Galois orbit size: $4$ Smallest permutation container: $C_8$ Parity: odd Dirichlet character: $$\chi_{160}(37,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{8} + 40 x^{6} + 500 x^{4} + 2000 x^{2} + 50$$  .

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 7.

Roots:
 $r_{ 1 }$ $=$ $$3 + 21\cdot 31 + 10\cdot 31^{2} + 22\cdot 31^{3} + 22\cdot 31^{4} + 17\cdot 31^{5} + 25\cdot 31^{6} +O(31^{7})$$ $r_{ 2 }$ $=$ $$8 + 21\cdot 31 + 8\cdot 31^{2} + 14\cdot 31^{3} + 30\cdot 31^{4} + 26\cdot 31^{5} + 13\cdot 31^{6} +O(31^{7})$$ $r_{ 3 }$ $=$ $$11 + 9\cdot 31 + 16\cdot 31^{2} + 26\cdot 31^{3} + 5\cdot 31^{4} + 28\cdot 31^{5} + 30\cdot 31^{6} +O(31^{7})$$ $r_{ 4 }$ $=$ $$13 + 23\cdot 31 + 24\cdot 31^{2} + 25\cdot 31^{3} + 27\cdot 31^{4} + 24\cdot 31^{5} +O(31^{7})$$ $r_{ 5 }$ $=$ $$18 + 7\cdot 31 + 6\cdot 31^{2} + 5\cdot 31^{3} + 3\cdot 31^{4} + 6\cdot 31^{5} + 30\cdot 31^{6} +O(31^{7})$$ $r_{ 6 }$ $=$ $$20 + 21\cdot 31 + 14\cdot 31^{2} + 4\cdot 31^{3} + 25\cdot 31^{4} + 2\cdot 31^{5} +O(31^{7})$$ $r_{ 7 }$ $=$ $$23 + 9\cdot 31 + 22\cdot 31^{2} + 16\cdot 31^{3} + 4\cdot 31^{5} + 17\cdot 31^{6} +O(31^{7})$$ $r_{ 8 }$ $=$ $$28 + 9\cdot 31 + 20\cdot 31^{2} + 8\cdot 31^{3} + 8\cdot 31^{4} + 13\cdot 31^{5} + 5\cdot 31^{6} +O(31^{7})$$

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $1$ $1$ $2$ $(1,8)(2,7)(3,6)(4,5)$ $-1$ $1$ $4$ $(1,7,8,2)(3,5,6,4)$ $-\zeta_{8}^{2}$ $1$ $4$ $(1,2,8,7)(3,4,6,5)$ $\zeta_{8}^{2}$ $1$ $8$ $(1,5,7,6,8,4,2,3)$ $\zeta_{8}^{3}$ $1$ $8$ $(1,6,2,5,8,3,7,4)$ $\zeta_{8}$ $1$ $8$ $(1,4,7,3,8,5,2,6)$ $-\zeta_{8}^{3}$ $1$ $8$ $(1,3,2,4,8,6,7,5)$ $-\zeta_{8}$

The blue line marks the conjugacy class containing complex conjugation.