# Properties

 Label 2.117.8t17.b.b Dimension $2$ Group $C_4\wr C_2$ Conductor $117$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $C_4\wr C_2$ Conductor: $$117$$$$\medspace = 3^{2} \cdot 13$$ Artin stem field: Galois closure of 8.0.1601613.1 Galois orbit size: $2$ Smallest permutation container: $C_4\wr C_2$ Parity: odd Determinant: 1.13.4t1.a.b Projective image: $D_4$ Projective stem field: Galois closure of 4.2.6591.1

## Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$48 + 52\cdot 139 + 37\cdot 139^{2} + 86\cdot 139^{3} + 30\cdot 139^{4} +O(139^{5})$$ 48 + 52*139 + 37*139^2 + 86*139^3 + 30*139^4+O(139^5) $r_{ 2 }$ $=$ $$55 + 24\cdot 139 + 70\cdot 139^{2} + 32\cdot 139^{3} + 93\cdot 139^{4} +O(139^{5})$$ 55 + 24*139 + 70*139^2 + 32*139^3 + 93*139^4+O(139^5) $r_{ 3 }$ $=$ $$74 + 93\cdot 139 + 61\cdot 139^{2} + 116\cdot 139^{3} + 51\cdot 139^{4} +O(139^{5})$$ 74 + 93*139 + 61*139^2 + 116*139^3 + 51*139^4+O(139^5) $r_{ 4 }$ $=$ $$77 + 83\cdot 139 + 15\cdot 139^{2} + 95\cdot 139^{3} + 98\cdot 139^{4} +O(139^{5})$$ 77 + 83*139 + 15*139^2 + 95*139^3 + 98*139^4+O(139^5) $r_{ 5 }$ $=$ $$90 + 116\cdot 139 + 20\cdot 139^{2} + 25\cdot 139^{3} + 118\cdot 139^{4} +O(139^{5})$$ 90 + 116*139 + 20*139^2 + 25*139^3 + 118*139^4+O(139^5) $r_{ 6 }$ $=$ $$93 + 94\cdot 139 + 130\cdot 139^{2} + 28\cdot 139^{3} + 62\cdot 139^{4} +O(139^{5})$$ 93 + 94*139 + 130*139^2 + 28*139^3 + 62*139^4+O(139^5) $r_{ 7 }$ $=$ $$122 + 72\cdot 139 + 124\cdot 139^{2} + 31\cdot 139^{3} + 123\cdot 139^{4} +O(139^{5})$$ 122 + 72*139 + 124*139^2 + 31*139^3 + 123*139^4+O(139^5) $r_{ 8 }$ $=$ $$136 + 17\cdot 139 + 95\cdot 139^{2} + 117\cdot 139^{4} +O(139^{5})$$ 136 + 17*139 + 95*139^2 + 117*139^4+O(139^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.