# Properties

 Label 2.95.8t6.a Dimension $2$ Group $D_{8}$ Conductor $95$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{8}$ Conductor: $$95$$$$\medspace = 5 \cdot 19$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 8.2.4286875.1 Galois orbit size: $2$ Smallest permutation container: $D_{8}$ Parity: odd Projective image: $D_4$ Projective field: Galois closure of 4.2.475.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$2 + 129\cdot 131 + 3\cdot 131^{2} + 118\cdot 131^{3} + 70\cdot 131^{4} +O(131^{5})$$ 2 + 129*131 + 3*131^2 + 118*131^3 + 70*131^4+O(131^5) $r_{ 2 }$ $=$ $$16 + 112\cdot 131 + 72\cdot 131^{2} + 124\cdot 131^{3} + 36\cdot 131^{4} +O(131^{5})$$ 16 + 112*131 + 72*131^2 + 124*131^3 + 36*131^4+O(131^5) $r_{ 3 }$ $=$ $$43 + 14\cdot 131 + 37\cdot 131^{2} + 104\cdot 131^{3} + 5\cdot 131^{4} +O(131^{5})$$ 43 + 14*131 + 37*131^2 + 104*131^3 + 5*131^4+O(131^5) $r_{ 4 }$ $=$ $$68 + 75\cdot 131 + 38\cdot 131^{2} + 30\cdot 131^{3} + 25\cdot 131^{4} +O(131^{5})$$ 68 + 75*131 + 38*131^2 + 30*131^3 + 25*131^4+O(131^5) $r_{ 5 }$ $=$ $$72 + 104\cdot 131 + 34\cdot 131^{2} + 104\cdot 131^{3} + 68\cdot 131^{4} +O(131^{5})$$ 72 + 104*131 + 34*131^2 + 104*131^3 + 68*131^4+O(131^5) $r_{ 6 }$ $=$ $$86 + 16\cdot 131 + 87\cdot 131^{2} + 94\cdot 131^{3} + 60\cdot 131^{4} +O(131^{5})$$ 86 + 16*131 + 87*131^2 + 94*131^3 + 60*131^4+O(131^5) $r_{ 7 }$ $=$ $$114 + 85\cdot 131 + 91\cdot 131^{2} + 84\cdot 131^{3} + 14\cdot 131^{4} +O(131^{5})$$ 114 + 85*131 + 91*131^2 + 84*131^3 + 14*131^4+O(131^5) $r_{ 8 }$ $=$ $$124 + 116\cdot 131 + 26\cdot 131^{2} + 125\cdot 131^{3} + 109\cdot 131^{4} +O(131^{5})$$ 124 + 116*131 + 26*131^2 + 125*131^3 + 109*131^4+O(131^5)

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $2$ $2$ $1$ $2$ $(1,7)(2,4)(3,8)(5,6)$ $-2$ $-2$ $4$ $2$ $(2,8)(3,4)(5,6)$ $0$ $0$ $4$ $2$ $(1,2)(3,5)(4,7)(6,8)$ $0$ $0$ $2$ $4$ $(1,6,7,5)(2,3,4,8)$ $0$ $0$ $2$ $8$ $(1,2,6,3,7,4,5,8)$ $-\zeta_{8}^{3} + \zeta_{8}$ $\zeta_{8}^{3} - \zeta_{8}$ $2$ $8$ $(1,3,5,2,7,8,6,4)$ $\zeta_{8}^{3} - \zeta_{8}$ $-\zeta_{8}^{3} + \zeta_{8}$
The blue line marks the conjugacy class containing complex conjugation.