# Properties

 Label 2.441.8t8.a.a Dimension $2$ Group $QD_{16}$ Conductor $441$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $QD_{16}$ Conductor: $$441$$$$\medspace = 3^{2} \cdot 7^{2}$$ Artin stem field: 8.2.257298363.1 Galois orbit size: $2$ Smallest permutation container: $QD_{16}$ Parity: odd Determinant: 1.3.2t1.a.a Projective image: $D_4$ Projective stem field: 4.0.189.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 2 x^{7} + x^{6} - 5 x^{5} + 7 x^{4} - 2 x^{3} + x^{2} - 5 x + 1$$  .

The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$5 + 39\cdot 79 + 56\cdot 79^{2} + 40\cdot 79^{3} + 4\cdot 79^{4} +O(79^{5})$$ $r_{ 2 }$ $=$ $$12 + 22\cdot 79 + 60\cdot 79^{2} + 46\cdot 79^{3} + 10\cdot 79^{4} +O(79^{5})$$ $r_{ 3 }$ $=$ $$13 + 13\cdot 79 + 3\cdot 79^{2} + 65\cdot 79^{3} + 28\cdot 79^{4} +O(79^{5})$$ $r_{ 4 }$ $=$ $$14 + 25\cdot 79 + 49\cdot 79^{2} + 17\cdot 79^{3} +O(79^{5})$$ $r_{ 5 }$ $=$ $$39 + 53\cdot 79 + 46\cdot 79^{2} + 48\cdot 79^{3} + 12\cdot 79^{4} +O(79^{5})$$ $r_{ 6 }$ $=$ $$49 + 71\cdot 79 + 70\cdot 79^{2} + 52\cdot 79^{3} + 63\cdot 79^{4} +O(79^{5})$$ $r_{ 7 }$ $=$ $$51 + 11\cdot 79 + 21\cdot 79^{2} + 34\cdot 79^{3} + 44\cdot 79^{4} +O(79^{5})$$ $r_{ 8 }$ $=$ $$56 + 8\cdot 79^{2} + 10\cdot 79^{3} + 72\cdot 79^{4} +O(79^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.