# Properties

 Label 2.1575.8t11.a.b Dimension $2$ Group $Q_8:C_2$ Conductor $1575$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8:C_2$ Conductor: $$1575$$$$\medspace = 3^{2} \cdot 5^{2} \cdot 7$$ Artin stem field: Galois closure of 8.0.558140625.1 Galois orbit size: $2$ Smallest permutation container: $Q_8:C_2$ Parity: odd Determinant: 1.35.2t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{-7})$$

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - x^{7} - 10x^{6} + 11x^{5} + 34x^{4} - 29x^{3} - 55x^{2} + 4x + 76$$ x^8 - x^7 - 10*x^6 + 11*x^5 + 34*x^4 - 29*x^3 - 55*x^2 + 4*x + 76 .

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

Roots:
 $r_{ 1 }$ $=$ $$22 + 82\cdot 109 + 67\cdot 109^{2} + 26\cdot 109^{4} +O(109^{5})$$ 22 + 82*109 + 67*109^2 + 26*109^4+O(109^5) $r_{ 2 }$ $=$ $$28 + 48\cdot 109 + 52\cdot 109^{2} + 88\cdot 109^{3} + 101\cdot 109^{4} +O(109^{5})$$ 28 + 48*109 + 52*109^2 + 88*109^3 + 101*109^4+O(109^5) $r_{ 3 }$ $=$ $$37 + 97\cdot 109 + 26\cdot 109^{2} + 84\cdot 109^{3} + 23\cdot 109^{4} +O(109^{5})$$ 37 + 97*109 + 26*109^2 + 84*109^3 + 23*109^4+O(109^5) $r_{ 4 }$ $=$ $$40 + 108\cdot 109 + 14\cdot 109^{2} + 80\cdot 109^{3} + 55\cdot 109^{4} +O(109^{5})$$ 40 + 108*109 + 14*109^2 + 80*109^3 + 55*109^4+O(109^5) $r_{ 5 }$ $=$ $$48 + 54\cdot 109 + 82\cdot 109^{2} + 7\cdot 109^{3} + 101\cdot 109^{4} +O(109^{5})$$ 48 + 54*109 + 82*109^2 + 7*109^3 + 101*109^4+O(109^5) $r_{ 6 }$ $=$ $$83 + 92\cdot 109 + 99\cdot 109^{2} + 3\cdot 109^{3} + 65\cdot 109^{4} +O(109^{5})$$ 83 + 92*109 + 99*109^2 + 3*109^3 + 65*109^4+O(109^5) $r_{ 7 }$ $=$ $$89 + 6\cdot 109 + 6\cdot 109^{2} + 68\cdot 109^{3} + 35\cdot 109^{4} +O(109^{5})$$ 89 + 6*109 + 6*109^2 + 68*109^3 + 35*109^4+O(109^5) $r_{ 8 }$ $=$ $$90 + 54\cdot 109 + 85\cdot 109^{2} + 102\cdot 109^{3} + 26\cdot 109^{4} +O(109^{5})$$ 90 + 54*109 + 85*109^2 + 102*109^3 + 26*109^4+O(109^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.