# Properties

 Label 2.1280.8t11.b.b Dimension $2$ Group $Q_8:C_2$ Conductor $1280$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8:C_2$ Conductor: $$1280$$$$\medspace = 2^{8} \cdot 5$$ Artin stem field: Galois closure of 8.0.419430400.3 Galois orbit size: $2$ Smallest permutation container: $Q_8:C_2$ Parity: odd Determinant: 1.40.2t1.b.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(i, \sqrt{5})$$

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 8x^{4} + 25$$ x^8 - 8*x^4 + 25 .

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 6.

Roots:
 $r_{ 1 }$ $=$ $$12 + 4\cdot 41 + 41^{2} + 40\cdot 41^{3} + 31\cdot 41^{4} + 9\cdot 41^{5} +O(41^{6})$$ 12 + 4*41 + 41^2 + 40*41^3 + 31*41^4 + 9*41^5+O(41^6) $r_{ 2 }$ $=$ $$15 + 17\cdot 41 + 35\cdot 41^{2} + 39\cdot 41^{3} + 38\cdot 41^{4} + 27\cdot 41^{5} +O(41^{6})$$ 15 + 17*41 + 35*41^2 + 39*41^3 + 38*41^4 + 27*41^5+O(41^6) $r_{ 3 }$ $=$ $$16 + 35\cdot 41 + 34\cdot 41^{2} + 32\cdot 41^{3} + 40\cdot 41^{4} + 36\cdot 41^{5} +O(41^{6})$$ 16 + 35*41 + 34*41^2 + 32*41^3 + 40*41^4 + 36*41^5+O(41^6) $r_{ 4 }$ $=$ $$20 + 29\cdot 41 + 12\cdot 41^{2} + 8\cdot 41^{3} + 23\cdot 41^{4} + 18\cdot 41^{5} +O(41^{6})$$ 20 + 29*41 + 12*41^2 + 8*41^3 + 23*41^4 + 18*41^5+O(41^6) $r_{ 5 }$ $=$ $$21 + 11\cdot 41 + 28\cdot 41^{2} + 32\cdot 41^{3} + 17\cdot 41^{4} + 22\cdot 41^{5} +O(41^{6})$$ 21 + 11*41 + 28*41^2 + 32*41^3 + 17*41^4 + 22*41^5+O(41^6) $r_{ 6 }$ $=$ $$25 + 5\cdot 41 + 6\cdot 41^{2} + 8\cdot 41^{3} + 4\cdot 41^{5} +O(41^{6})$$ 25 + 5*41 + 6*41^2 + 8*41^3 + 4*41^5+O(41^6) $r_{ 7 }$ $=$ $$26 + 23\cdot 41 + 5\cdot 41^{2} + 41^{3} + 2\cdot 41^{4} + 13\cdot 41^{5} +O(41^{6})$$ 26 + 23*41 + 5*41^2 + 41^3 + 2*41^4 + 13*41^5+O(41^6) $r_{ 8 }$ $=$ $$29 + 36\cdot 41 + 39\cdot 41^{2} + 9\cdot 41^{4} + 31\cdot 41^{5} +O(41^{6})$$ 29 + 36*41 + 39*41^2 + 9*41^4 + 31*41^5+O(41^6)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.