# Properties

 Label 1.160.8t1.c.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: 8.8.33554432000000.2 Galois orbit size: $4$ Smallest permutation container: $C_8$ Parity: even Dirichlet character: $$\chi_{160}(43,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 40 x^{6} + 500 x^{4} - 2000 x^{2} + 2450$$  .

The roots of $f$ are computed in $\Q_{ 73 }$ to precision 6.

Roots:
 $r_{ 1 }$ $=$ $$2 + 34\cdot 73 + 14\cdot 73^{2} + 21\cdot 73^{3} + 8\cdot 73^{4} + 18\cdot 73^{5} +O(73^{6})$$ $r_{ 2 }$ $=$ $$4 + 56\cdot 73 + 3\cdot 73^{2} + 40\cdot 73^{3} + 41\cdot 73^{4} + 10\cdot 73^{5} +O(73^{6})$$ $r_{ 3 }$ $=$ $$23 + 10\cdot 73 + 4\cdot 73^{2} + 20\cdot 73^{3} + 13\cdot 73^{4} + 66\cdot 73^{5} +O(73^{6})$$ $r_{ 4 }$ $=$ $$32 + 21\cdot 73 + 59\cdot 73^{2} + 4\cdot 73^{3} + 21\cdot 73^{4} + 57\cdot 73^{5} +O(73^{6})$$ $r_{ 5 }$ $=$ $$41 + 51\cdot 73 + 13\cdot 73^{2} + 68\cdot 73^{3} + 51\cdot 73^{4} + 15\cdot 73^{5} +O(73^{6})$$ $r_{ 6 }$ $=$ $$50 + 62\cdot 73 + 68\cdot 73^{2} + 52\cdot 73^{3} + 59\cdot 73^{4} + 6\cdot 73^{5} +O(73^{6})$$ $r_{ 7 }$ $=$ $$69 + 16\cdot 73 + 69\cdot 73^{2} + 32\cdot 73^{3} + 31\cdot 73^{4} + 62\cdot 73^{5} +O(73^{6})$$ $r_{ 8 }$ $=$ $$71 + 38\cdot 73 + 58\cdot 73^{2} + 51\cdot 73^{3} + 64\cdot 73^{4} + 54\cdot 73^{5} +O(73^{6})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.