# Properties

 Label 2.192.8t11.b.b Dimension $2$ Group $Q_8:C_2$ Conductor $192$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8:C_2$ Conductor: $$192$$$$\medspace = 2^{6} \cdot 3$$ Artin stem field: 8.0.5308416.2 Galois orbit size: $2$ Smallest permutation container: $Q_8:C_2$ Parity: odd Determinant: 1.24.2t1.b.a Projective image: $C_2^2$ Projective field: $$\Q(\sqrt{-2}, \sqrt{-3})$$

## Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$6 + 32\cdot 73 + 55\cdot 73^{2} + 61\cdot 73^{3} + 29\cdot 73^{4} +O(73^{5})$$ $r_{ 2 }$ $=$ $$26 + 58\cdot 73 + 44\cdot 73^{3} + 46\cdot 73^{4} +O(73^{5})$$ $r_{ 3 }$ $=$ $$29 + 41\cdot 73 + 3\cdot 73^{2} + 16\cdot 73^{3} + 38\cdot 73^{4} +O(73^{5})$$ $r_{ 4 }$ $=$ $$36 + 50\cdot 73 + 42\cdot 73^{2} + 41\cdot 73^{3} + 59\cdot 73^{4} +O(73^{5})$$ $r_{ 5 }$ $=$ $$37 + 22\cdot 73 + 30\cdot 73^{2} + 31\cdot 73^{3} + 13\cdot 73^{4} +O(73^{5})$$ $r_{ 6 }$ $=$ $$44 + 31\cdot 73 + 69\cdot 73^{2} + 56\cdot 73^{3} + 34\cdot 73^{4} +O(73^{5})$$ $r_{ 7 }$ $=$ $$47 + 14\cdot 73 + 72\cdot 73^{2} + 28\cdot 73^{3} + 26\cdot 73^{4} +O(73^{5})$$ $r_{ 8 }$ $=$ $$67 + 40\cdot 73 + 17\cdot 73^{2} + 11\cdot 73^{3} + 43\cdot 73^{4} +O(73^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.