# Properties

 Label 2.260.8t17.d.a Dimension $2$ Group $C_4\wr C_2$ Conductor $260$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $C_4\wr C_2$ Conductor: $$260$$$$\medspace = 2^{2} \cdot 5 \cdot 13$$ Artin stem field: Galois closure of 8.0.70304000.3 Galois orbit size: $2$ Smallest permutation container: $C_4\wr C_2$ Parity: odd Determinant: 1.260.4t1.c.a Projective image: $D_4$ Projective stem field: Galois closure of 4.2.1098500.1

## Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 73 }$ to precision 6.

Roots:
 $r_{ 1 }$ $=$ $$10 + 34\cdot 73 + 23\cdot 73^{2} + 42\cdot 73^{3} + 28\cdot 73^{4} + 45\cdot 73^{5} +O(73^{6})$$ 10 + 34*73 + 23*73^2 + 42*73^3 + 28*73^4 + 45*73^5+O(73^6) $r_{ 2 }$ $=$ $$22 + 49\cdot 73 + 22\cdot 73^{2} + 38\cdot 73^{3} + 73^{4} + 27\cdot 73^{5} +O(73^{6})$$ 22 + 49*73 + 22*73^2 + 38*73^3 + 73^4 + 27*73^5+O(73^6) $r_{ 3 }$ $=$ $$23 + 64\cdot 73 + 25\cdot 73^{2} + 73^{3} + 70\cdot 73^{4} + 56\cdot 73^{5} +O(73^{6})$$ 23 + 64*73 + 25*73^2 + 73^3 + 70*73^4 + 56*73^5+O(73^6) $r_{ 4 }$ $=$ $$28 + 7\cdot 73 + 65\cdot 73^{2} + 45\cdot 73^{3} + 48\cdot 73^{4} + 58\cdot 73^{5} +O(73^{6})$$ 28 + 7*73 + 65*73^2 + 45*73^3 + 48*73^4 + 58*73^5+O(73^6) $r_{ 5 }$ $=$ $$29 + 36\cdot 73 + 55\cdot 73^{2} + 47\cdot 73^{3} + 57\cdot 73^{4} + 19\cdot 73^{5} +O(73^{6})$$ 29 + 36*73 + 55*73^2 + 47*73^3 + 57*73^4 + 19*73^5+O(73^6) $r_{ 6 }$ $=$ $$54 + 68\cdot 73 + 3\cdot 73^{2} + 23\cdot 73^{3} + 21\cdot 73^{4} + 22\cdot 73^{5} +O(73^{6})$$ 54 + 68*73 + 3*73^2 + 23*73^3 + 21*73^4 + 22*73^5+O(73^6) $r_{ 7 }$ $=$ $$60 + 65\cdot 73 + 5\cdot 73^{2} + 36\cdot 73^{3} + 8\cdot 73^{4} + 35\cdot 73^{5} +O(73^{6})$$ 60 + 65*73 + 5*73^2 + 36*73^3 + 8*73^4 + 35*73^5+O(73^6) $r_{ 8 }$ $=$ $$68 + 38\cdot 73 + 16\cdot 73^{2} + 57\cdot 73^{3} + 55\cdot 73^{4} + 26\cdot 73^{5} +O(73^{6})$$ 68 + 38*73 + 16*73^2 + 57*73^3 + 55*73^4 + 26*73^5+O(73^6)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.