# Properties

 Label 1.32.8t1.a.a Dimension $1$ Group $C_8$ Conductor $32$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_8$ Conductor: $$32$$$$\medspace = 2^{5}$$ Artin field: Galois closure of $$\Q(\zeta_{32})^+$$ Galois orbit size: $4$ Smallest permutation container: $C_8$ Parity: even Dirichlet character: $$\chi_{32}(13,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 8x^{6} + 20x^{4} - 16x^{2} + 2$$ x^8 - 8*x^6 + 20*x^4 - 16*x^2 + 2 .

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.