# Properties

 Label 2.273.8t17.d.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 8.0.8719893.2 Galois orbit size: $2$ Smallest permutation container: $C_4\wr C_2$ Parity: odd Determinant: 1.273.4t1.a.a Projective image: $D_4$ Projective stem field: Galois closure of 4.2.322959.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 3x^{7} + 5x^{5} + x^{4} - 3x^{3} - 5x^{2} - 2x + 7$$ x^8 - 3*x^7 + 5*x^5 + x^4 - 3*x^3 - 5*x^2 - 2*x + 7 .

The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$7 + 97\cdot 103 + 86\cdot 103^{2} + 66\cdot 103^{3} + 43\cdot 103^{4} +O(103^{5})$$ 7 + 97*103 + 86*103^2 + 66*103^3 + 43*103^4+O(103^5) $r_{ 2 }$ $=$ $$10 + 51\cdot 103 + 32\cdot 103^{2} + 35\cdot 103^{3} + 79\cdot 103^{4} +O(103^{5})$$ 10 + 51*103 + 32*103^2 + 35*103^3 + 79*103^4+O(103^5) $r_{ 3 }$ $=$ $$18 + 34\cdot 103 + 62\cdot 103^{2} + 66\cdot 103^{3} + 55\cdot 103^{4} +O(103^{5})$$ 18 + 34*103 + 62*103^2 + 66*103^3 + 55*103^4+O(103^5) $r_{ 4 }$ $=$ $$19 + 51\cdot 103 + 83\cdot 103^{2} + 89\cdot 103^{3} + 43\cdot 103^{4} +O(103^{5})$$ 19 + 51*103 + 83*103^2 + 89*103^3 + 43*103^4+O(103^5) $r_{ 5 }$ $=$ $$41 + 98\cdot 103 + 68\cdot 103^{2} + 28\cdot 103^{3} + 100\cdot 103^{4} +O(103^{5})$$ 41 + 98*103 + 68*103^2 + 28*103^3 + 100*103^4+O(103^5) $r_{ 6 }$ $=$ $$44 + 2\cdot 103 + 14\cdot 103^{2} + 22\cdot 103^{3} + 6\cdot 103^{4} +O(103^{5})$$ 44 + 2*103 + 14*103^2 + 22*103^3 + 6*103^4+O(103^5) $r_{ 7 }$ $=$ $$81 + 38\cdot 103 + 102\cdot 103^{2} + 80\cdot 103^{3} + 17\cdot 103^{4} +O(103^{5})$$ 81 + 38*103 + 102*103^2 + 80*103^3 + 17*103^4+O(103^5) $r_{ 8 }$ $=$ $$92 + 38\cdot 103 + 64\cdot 103^{2} + 21\cdot 103^{3} + 65\cdot 103^{4} +O(103^{5})$$ 92 + 38*103 + 64*103^2 + 21*103^3 + 65*103^4+O(103^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.