# Properties

 Label 2.1792.8t8.b.b Dimension $2$ Group $QD_{16}$ Conductor $1792$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $QD_{16}$ Conductor: $$1792$$$$\medspace = 2^{8} \cdot 7$$ Artin stem field: 8.2.5754585088.3 Galois orbit size: $2$ Smallest permutation container: $QD_{16}$ Parity: odd Determinant: 1.7.2t1.a.a Projective image: $D_4$ Projective stem field: 4.0.1568.1

## Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 71 }$ to precision 8.

Roots:
 $r_{ 1 }$ $=$ $$5 + 35\cdot 71 + 22\cdot 71^{2} + 64\cdot 71^{3} + 68\cdot 71^{4} + 71^{5} + 69\cdot 71^{6} + 27\cdot 71^{7} +O(71^{8})$$ $r_{ 2 }$ $=$ $$6 + 60\cdot 71 + 71^{2} + 29\cdot 71^{3} + 48\cdot 71^{4} + 41\cdot 71^{5} + 58\cdot 71^{6} + 16\cdot 71^{7} +O(71^{8})$$ $r_{ 3 }$ $=$ $$18 + 4\cdot 71 + 70\cdot 71^{2} + 38\cdot 71^{3} + 62\cdot 71^{4} + 40\cdot 71^{5} + 18\cdot 71^{6} + 60\cdot 71^{7} +O(71^{8})$$ $r_{ 4 }$ $=$ $$20 + 67\cdot 71 + 64\cdot 71^{2} + 37\cdot 71^{3} + 50\cdot 71^{4} + 6\cdot 71^{5} + 49\cdot 71^{6} + 24\cdot 71^{7} +O(71^{8})$$ $r_{ 5 }$ $=$ $$51 + 3\cdot 71 + 6\cdot 71^{2} + 33\cdot 71^{3} + 20\cdot 71^{4} + 64\cdot 71^{5} + 21\cdot 71^{6} + 46\cdot 71^{7} +O(71^{8})$$ $r_{ 6 }$ $=$ $$53 + 66\cdot 71 + 32\cdot 71^{3} + 8\cdot 71^{4} + 30\cdot 71^{5} + 52\cdot 71^{6} + 10\cdot 71^{7} +O(71^{8})$$ $r_{ 7 }$ $=$ $$65 + 10\cdot 71 + 69\cdot 71^{2} + 41\cdot 71^{3} + 22\cdot 71^{4} + 29\cdot 71^{5} + 12\cdot 71^{6} + 54\cdot 71^{7} +O(71^{8})$$ $r_{ 8 }$ $=$ $$66 + 35\cdot 71 + 48\cdot 71^{2} + 6\cdot 71^{3} + 2\cdot 71^{4} + 69\cdot 71^{5} + 71^{6} + 43\cdot 71^{7} +O(71^{8})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.