# Properties

 Label 2.180.8t11.b Dimension $2$ Group $Q_8:C_2$ Conductor $180$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8:C_2$ Conductor: $$180$$$$\medspace = 2^{2} \cdot 3^{2} \cdot 5$$ Artin number field: Galois closure of 8.0.4665600.1 Galois orbit size: $2$ Smallest permutation container: $Q_8:C_2$ Parity: odd Projective image: $C_2^2$ Projective field: $$\Q(i, \sqrt{15})$$

## Galois action

### Roots of defining polynomial

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 values $c1$ $c2$ $1$ $1$ $()$ $2$ $2$ $1$ $2$ $(1,4)(2,5)(3,8)(6,7)$ $-2$ $-2$ $2$ $2$ $(1,5)(2,4)(3,7)(6,8)$ $0$ $0$ $2$ $2$ $(2,5)(6,7)$ $0$ $0$ $2$ $2$ $(1,6)(2,8)(3,5)(4,7)$ $0$ $0$ $1$ $4$ $(1,3,4,8)(2,6,5,7)$ $-2 \zeta_{4}$ $2 \zeta_{4}$ $1$ $4$ $(1,8,4,3)(2,7,5,6)$ $2 \zeta_{4}$ $-2 \zeta_{4}$ $2$ $4$ $(1,6,4,7)(2,3,5,8)$ $0$ $0$ $2$ $4$ $(1,3,4,8)(2,7,5,6)$ $0$ $0$ $2$ $4$ $(1,5,4,2)(3,7,8,6)$ $0$ $0$
The blue line marks the conjugacy class containing complex conjugation.