# Properties

 Label 2.273.8t17.c.a Dimension $2$ Group $C_4\wr C_2$ Conductor $273$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $C_4\wr C_2$ Conductor: $$273$$$$\medspace = 3 \cdot 7 \cdot 13$$ Artin stem field: Galois closure of 8.0.8719893.1 Galois orbit size: $2$ Smallest permutation container: $C_4\wr C_2$ Parity: odd Determinant: 1.273.4t1.a.b Projective image: $D_4$ Projective stem field: Galois closure of 4.2.322959.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 3x^{7} - x^{6} + 11x^{5} - 6x^{4} - 10x^{3} + 9x^{2} - x + 1$$ x^8 - 3*x^7 - x^6 + 11*x^5 - 6*x^4 - 10*x^3 + 9*x^2 - x + 1 .

The roots of $f$ are computed in $\Q_{ 181 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$3 + 22\cdot 181 + 119\cdot 181^{2} + 127\cdot 181^{3} + 146\cdot 181^{4} +O(181^{5})$$ 3 + 22*181 + 119*181^2 + 127*181^3 + 146*181^4+O(181^5) $r_{ 2 }$ $=$ $$10 + 109\cdot 181 + 174\cdot 181^{2} + 179\cdot 181^{3} + 64\cdot 181^{4} +O(181^{5})$$ 10 + 109*181 + 174*181^2 + 179*181^3 + 64*181^4+O(181^5) $r_{ 3 }$ $=$ $$36 + 155\cdot 181 + 70\cdot 181^{2} + 151\cdot 181^{3} + 124\cdot 181^{4} +O(181^{5})$$ 36 + 155*181 + 70*181^2 + 151*181^3 + 124*181^4+O(181^5) $r_{ 4 }$ $=$ $$42 + 145\cdot 181 + 181^{2} + 36\cdot 181^{3} + 118\cdot 181^{4} +O(181^{5})$$ 42 + 145*181 + 181^2 + 36*181^3 + 118*181^4+O(181^5) $r_{ 5 }$ $=$ $$67 + 89\cdot 181 + 109\cdot 181^{2} + 58\cdot 181^{3} + 121\cdot 181^{4} +O(181^{5})$$ 67 + 89*181 + 109*181^2 + 58*181^3 + 121*181^4+O(181^5) $r_{ 6 }$ $=$ $$119 + 103\cdot 181 + 47\cdot 181^{2} + 136\cdot 181^{3} + 85\cdot 181^{4} +O(181^{5})$$ 119 + 103*181 + 47*181^2 + 136*181^3 + 85*181^4+O(181^5) $r_{ 7 }$ $=$ $$127 + 82\cdot 181 + 52\cdot 181^{2} + 106\cdot 181^{3} + 157\cdot 181^{4} +O(181^{5})$$ 127 + 82*181 + 52*181^2 + 106*181^3 + 157*181^4+O(181^5) $r_{ 8 }$ $=$ $$142 + 16\cdot 181 + 148\cdot 181^{2} + 108\cdot 181^{3} + 85\cdot 181^{4} +O(181^{5})$$ 142 + 16*181 + 148*181^2 + 108*181^3 + 85*181^4+O(181^5)

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,5)(2,7)(3,8)(4,6)$ $-2$ $2$ $2$ $(1,5)(4,6)$ $0$ $4$ $2$ $(1,8)(2,4)(3,5)(6,7)$ $0$ $1$ $4$ $(1,6,5,4)(2,8,7,3)$ $2 \zeta_{4}$ $1$ $4$ $(1,4,5,6)(2,3,7,8)$ $-2 \zeta_{4}$ $2$ $4$ $(1,6,5,4)(2,3,7,8)$ $0$ $2$ $4$ $(1,4,5,6)$ $-\zeta_{4} + 1$ $2$ $4$ $(1,6,5,4)$ $\zeta_{4} + 1$ $2$ $4$ $(1,6,5,4)(2,7)(3,8)$ $\zeta_{4} - 1$ $2$ $4$ $(1,4,5,6)(2,7)(3,8)$ $-\zeta_{4} - 1$ $4$ $4$ $(1,7,5,2)(3,4,8,6)$ $0$ $4$ $8$ $(1,7,6,3,5,2,4,8)$ $0$ $4$ $8$ $(1,3,4,7,5,8,6,2)$ $0$

The blue line marks the conjugacy class containing complex conjugation.