# Properties

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

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## Basic invariants

 Dimension: $1$ Group: $C_8$ Conductor: $$160$$$$\medspace = 2^{5} \cdot 5$$ Artin field: 8.8.1342177280000.1 Galois orbit size: $4$ Smallest permutation container: $C_8$ Parity: even Dirichlet character: $$\chi_{160}(109,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 40 x^{6} + 500 x^{4} - 2000 x^{2} + 1250$$  .

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

Roots:
 $r_{ 1 }$ $=$ $$2 + 20\cdot 31 + 9\cdot 31^{2} + 25\cdot 31^{3} + 3\cdot 31^{4} + 20\cdot 31^{5} + 12\cdot 31^{6} +O(31^{7})$$ $r_{ 2 }$ $=$ $$4 + 21\cdot 31 + 25\cdot 31^{2} + 20\cdot 31^{3} + 26\cdot 31^{4} + 14\cdot 31^{5} + 15\cdot 31^{6} +O(31^{7})$$ $r_{ 3 }$ $=$ $$7 + 2\cdot 31 + 25\cdot 31^{2} + 17\cdot 31^{3} + 14\cdot 31^{4} + 23\cdot 31^{5} + 8\cdot 31^{6} +O(31^{7})$$ $r_{ 4 }$ $=$ $$8 + 3\cdot 31^{2} + 11\cdot 31^{3} + 30\cdot 31^{4} + 19\cdot 31^{5} + 6\cdot 31^{6} +O(31^{7})$$ $r_{ 5 }$ $=$ $$23 + 30\cdot 31 + 27\cdot 31^{2} + 19\cdot 31^{3} + 11\cdot 31^{5} + 24\cdot 31^{6} +O(31^{7})$$ $r_{ 6 }$ $=$ $$24 + 28\cdot 31 + 5\cdot 31^{2} + 13\cdot 31^{3} + 16\cdot 31^{4} + 7\cdot 31^{5} + 22\cdot 31^{6} +O(31^{7})$$ $r_{ 7 }$ $=$ $$27 + 9\cdot 31 + 5\cdot 31^{2} + 10\cdot 31^{3} + 4\cdot 31^{4} + 16\cdot 31^{5} + 15\cdot 31^{6} +O(31^{7})$$ $r_{ 8 }$ $=$ $$29 + 10\cdot 31 + 21\cdot 31^{2} + 5\cdot 31^{3} + 27\cdot 31^{4} + 10\cdot 31^{5} + 18\cdot 31^{6} +O(31^{7})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.