# Properties

 Label 2.1280.8t11.a.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 8.0.419430400.2 Galois orbit size: $2$ Smallest permutation container: $Q_8:C_2$ Parity: odd Determinant: 1.20.2t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(i, \sqrt{10})$$

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 12x^{6} + 48x^{4} - 60x^{2} + 25$$ x^8 - 12*x^6 + 48*x^4 - 60*x^2 + 25 .

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

Roots:
 $r_{ 1 }$ $=$ $$2 + 23\cdot 41 + 3\cdot 41^{2} + 12\cdot 41^{3} + 38\cdot 41^{4} + 10\cdot 41^{5} +O(41^{6})$$ 2 + 23*41 + 3*41^2 + 12*41^3 + 38*41^4 + 10*41^5+O(41^6) $r_{ 2 }$ $=$ $$5 + 34\cdot 41 + 11\cdot 41^{2} + 37\cdot 41^{3} + 34\cdot 41^{4} + 13\cdot 41^{5} +O(41^{6})$$ 5 + 34*41 + 11*41^2 + 37*41^3 + 34*41^4 + 13*41^5+O(41^6) $r_{ 3 }$ $=$ $$14 + 41 + 26\cdot 41^{2} + 34\cdot 41^{3} + 18\cdot 41^{4} + 10\cdot 41^{5} +O(41^{6})$$ 14 + 41 + 26*41^2 + 34*41^3 + 18*41^4 + 10*41^5+O(41^6) $r_{ 4 }$ $=$ $$19 + 35\cdot 41 + 11\cdot 41^{2} + 8\cdot 41^{3} + 29\cdot 41^{4} + 20\cdot 41^{5} +O(41^{6})$$ 19 + 35*41 + 11*41^2 + 8*41^3 + 29*41^4 + 20*41^5+O(41^6) $r_{ 5 }$ $=$ $$22 + 5\cdot 41 + 29\cdot 41^{2} + 32\cdot 41^{3} + 11\cdot 41^{4} + 20\cdot 41^{5} +O(41^{6})$$ 22 + 5*41 + 29*41^2 + 32*41^3 + 11*41^4 + 20*41^5+O(41^6) $r_{ 6 }$ $=$ $$27 + 39\cdot 41 + 14\cdot 41^{2} + 6\cdot 41^{3} + 22\cdot 41^{4} + 30\cdot 41^{5} +O(41^{6})$$ 27 + 39*41 + 14*41^2 + 6*41^3 + 22*41^4 + 30*41^5+O(41^6) $r_{ 7 }$ $=$ $$36 + 6\cdot 41 + 29\cdot 41^{2} + 3\cdot 41^{3} + 6\cdot 41^{4} + 27\cdot 41^{5} +O(41^{6})$$ 36 + 6*41 + 29*41^2 + 3*41^3 + 6*41^4 + 27*41^5+O(41^6) $r_{ 8 }$ $=$ $$39 + 17\cdot 41 + 37\cdot 41^{2} + 28\cdot 41^{3} + 2\cdot 41^{4} + 30\cdot 41^{5} +O(41^{6})$$ 39 + 17*41 + 37*41^2 + 28*41^3 + 2*41^4 + 30*41^5+O(41^6)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.