# Properties

 Label 2.180.8t11.b.b Dimension $2$ Group $Q_8:C_2$ Conductor $180$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8:C_2$ Conductor: $$180$$$$\medspace = 2^{2} \cdot 3^{2} \cdot 5$$ Artin stem field: 8.0.4665600.1 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: $$\Q(i, \sqrt{15})$$

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 2 x^{7} + 2 x^{6} + 4 x^{5} - 8 x^{4} + 2 x^{3} + 5 x^{2} - 4 x + 1$$  .

The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$5 + 84\cdot 109 + 49\cdot 109^{2} + 42\cdot 109^{3} + 38\cdot 109^{4} +O(109^{5})$$ $r_{ 2 }$ $=$ $$14 + 63\cdot 109 + 17\cdot 109^{2} + 87\cdot 109^{3} + 75\cdot 109^{4} +O(109^{5})$$ $r_{ 3 }$ $=$ $$15 + 81\cdot 109 + 92\cdot 109^{2} + 9\cdot 109^{3} + 93\cdot 109^{4} +O(109^{5})$$ $r_{ 4 }$ $=$ $$67 + 2\cdot 109 + 59\cdot 109^{2} + 90\cdot 109^{3} + 10\cdot 109^{4} +O(109^{5})$$ $r_{ 5 }$ $=$ $$73 + 70\cdot 109 + 42\cdot 109^{2} + 104\cdot 109^{3} + 2\cdot 109^{4} +O(109^{5})$$ $r_{ 6 }$ $=$ $$75 + 102\cdot 109 + 35\cdot 109^{2} + 67\cdot 109^{3} + 85\cdot 109^{4} +O(109^{5})$$ $r_{ 7 }$ $=$ $$90 + 37\cdot 109 + 107\cdot 109^{2} + 36\cdot 109^{3} + 35\cdot 109^{4} +O(109^{5})$$ $r_{ 8 }$ $=$ $$99 + 102\cdot 109 + 30\cdot 109^{2} + 106\cdot 109^{3} + 93\cdot 109^{4} +O(109^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.