# Properties

 Label 2.1560.8t11.f.b Dimension $2$ Group $Q_8:C_2$ Conductor $1560$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8:C_2$ Conductor: $$1560$$$$\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 13$$ Artin stem field: Galois closure of 8.0.3701505600.2 Galois orbit size: $2$ Smallest permutation container: $Q_8:C_2$ Parity: odd Determinant: 1.1560.2t1.b.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{-30}, \sqrt{-39})$$

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 12x^{6} - 21x^{5} + 23x^{4} + 126x^{3} + 198x^{2} + 156x + 52$$ x^8 - 12*x^6 - 21*x^5 + 23*x^4 + 126*x^3 + 198*x^2 + 156*x + 52 .

The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$12 + 95\cdot 157 + 120\cdot 157^{2} + 75\cdot 157^{3} + 70\cdot 157^{4} +O(157^{5})$$ 12 + 95*157 + 120*157^2 + 75*157^3 + 70*157^4+O(157^5) $r_{ 2 }$ $=$ $$25 + 87\cdot 157 + 98\cdot 157^{2} + 13\cdot 157^{3} + 149\cdot 157^{4} +O(157^{5})$$ 25 + 87*157 + 98*157^2 + 13*157^3 + 149*157^4+O(157^5) $r_{ 3 }$ $=$ $$55 + 63\cdot 157 + 147\cdot 157^{2} + 84\cdot 157^{3} + 100\cdot 157^{4} +O(157^{5})$$ 55 + 63*157 + 147*157^2 + 84*157^3 + 100*157^4+O(157^5) $r_{ 4 }$ $=$ $$65 + 68\cdot 157 + 104\cdot 157^{2} + 139\cdot 157^{3} + 150\cdot 157^{4} +O(157^{5})$$ 65 + 68*157 + 104*157^2 + 139*157^3 + 150*157^4+O(157^5) $r_{ 5 }$ $=$ $$96 + 42\cdot 157 + 143\cdot 157^{2} + 4\cdot 157^{3} + 41\cdot 157^{4} +O(157^{5})$$ 96 + 42*157 + 143*157^2 + 4*157^3 + 41*157^4+O(157^5) $r_{ 6 }$ $=$ $$106 + 102\cdot 157 + 82\cdot 157^{2} + 126\cdot 157^{3} + 64\cdot 157^{4} +O(157^{5})$$ 106 + 102*157 + 82*157^2 + 126*157^3 + 64*157^4+O(157^5) $r_{ 7 }$ $=$ $$116 + 38\cdot 157 + 138\cdot 157^{2} + 16\cdot 157^{3} + 148\cdot 157^{4} +O(157^{5})$$ 116 + 38*157 + 138*157^2 + 16*157^3 + 148*157^4+O(157^5) $r_{ 8 }$ $=$ $$153 + 129\cdot 157 + 106\cdot 157^{2} + 8\cdot 157^{3} + 60\cdot 157^{4} +O(157^{5})$$ 153 + 129*157 + 106*157^2 + 8*157^3 + 60*157^4+O(157^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.