# Properties

 Label 2.3920.8t11.c.a Dimension $2$ Group $Q_8:C_2$ Conductor $3920$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8:C_2$ Conductor: $$3920$$$$\medspace = 2^{4} \cdot 5 \cdot 7^{2}$$ Artin stem field: 8.0.12047257600.1 Galois orbit size: $2$ Smallest permutation container: $Q_8:C_2$ Parity: odd Determinant: 1.20.2t1.a.a Projective image: $C_2^2$ Projective field: $$\Q(i, \sqrt{35})$$

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 4 x^{7} + 8 x^{6} - 10 x^{5} + 19 x^{4} + 2 x^{3} - 40 x^{2} + 24 x + 74$$  .

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

Roots:
 $r_{ 1 }$ $=$ $$2 + 70\cdot 109 + 98\cdot 109^{2} + 57\cdot 109^{3} + 46\cdot 109^{4} +O(109^{5})$$ $r_{ 2 }$ $=$ $$6 + 104\cdot 109^{2} + 41\cdot 109^{3} + 49\cdot 109^{4} +O(109^{5})$$ $r_{ 3 }$ $=$ $$28 + 56\cdot 109 + 99\cdot 109^{2} + 35\cdot 109^{3} + 41\cdot 109^{4} +O(109^{5})$$ $r_{ 4 }$ $=$ $$33 + 77\cdot 109 + 75\cdot 109^{2} + 31\cdot 109^{3} + 20\cdot 109^{4} +O(109^{5})$$ $r_{ 5 }$ $=$ $$44 + 84\cdot 109 + 47\cdot 109^{2} + 108\cdot 109^{3} + 106\cdot 109^{4} +O(109^{5})$$ $r_{ 6 }$ $=$ $$53 + 45\cdot 109 + 19\cdot 109^{2} + 32\cdot 109^{3} + 70\cdot 109^{4} +O(109^{5})$$ $r_{ 7 }$ $=$ $$75 + 91\cdot 109 + 24\cdot 109^{2} + 82\cdot 109^{3} + 80\cdot 109^{4} +O(109^{5})$$ $r_{ 8 }$ $=$ $$90 + 10\cdot 109 + 75\cdot 109^{2} + 45\cdot 109^{3} + 20\cdot 109^{4} +O(109^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.