# Properties

 Label 2.768.8t11.c.b Dimension $2$ Group $Q_8:C_2$ Conductor $768$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8:C_2$ Conductor: $$768$$$$\medspace = 2^{8} \cdot 3$$ Artin stem field: Galois closure of 8.0.150994944.1 Galois orbit size: $2$ Smallest permutation container: $Q_8:C_2$ Parity: odd Determinant: 1.3.2t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{-2}, \sqrt{3})$$

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 4x^{6} + 8x^{4} - 12x^{2} + 9$$ x^8 - 4*x^6 + 8*x^4 - 12*x^2 + 9 .

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

Roots:
 $r_{ 1 }$ $=$ $$14 + 55\cdot 73 + 69\cdot 73^{2} + 53\cdot 73^{3} + 25\cdot 73^{4} +O(73^{5})$$ 14 + 55*73 + 69*73^2 + 53*73^3 + 25*73^4+O(73^5) $r_{ 2 }$ $=$ $$15 + 67\cdot 73 + 48\cdot 73^{2} + 44\cdot 73^{3} + 57\cdot 73^{4} +O(73^{5})$$ 15 + 67*73 + 48*73^2 + 44*73^3 + 57*73^4+O(73^5) $r_{ 3 }$ $=$ $$16 + 34\cdot 73 + 19\cdot 73^{2} + 40\cdot 73^{3} + 70\cdot 73^{4} +O(73^{5})$$ 16 + 34*73 + 19*73^2 + 40*73^3 + 70*73^4+O(73^5) $r_{ 4 }$ $=$ $$35 + 46\cdot 73 + 67\cdot 73^{2} + 69\cdot 73^{3} + 37\cdot 73^{4} +O(73^{5})$$ 35 + 46*73 + 67*73^2 + 69*73^3 + 37*73^4+O(73^5) $r_{ 5 }$ $=$ $$38 + 26\cdot 73 + 5\cdot 73^{2} + 3\cdot 73^{3} + 35\cdot 73^{4} +O(73^{5})$$ 38 + 26*73 + 5*73^2 + 3*73^3 + 35*73^4+O(73^5) $r_{ 6 }$ $=$ $$57 + 38\cdot 73 + 53\cdot 73^{2} + 32\cdot 73^{3} + 2\cdot 73^{4} +O(73^{5})$$ 57 + 38*73 + 53*73^2 + 32*73^3 + 2*73^4+O(73^5) $r_{ 7 }$ $=$ $$58 + 5\cdot 73 + 24\cdot 73^{2} + 28\cdot 73^{3} + 15\cdot 73^{4} +O(73^{5})$$ 58 + 5*73 + 24*73^2 + 28*73^3 + 15*73^4+O(73^5) $r_{ 8 }$ $=$ $$59 + 17\cdot 73 + 3\cdot 73^{2} + 19\cdot 73^{3} + 47\cdot 73^{4} +O(73^{5})$$ 59 + 17*73 + 3*73^2 + 19*73^3 + 47*73^4+O(73^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.