# Properties

 Label 2.896.8t6.a.b Dimension $2$ Group $D_{8}$ Conductor $896$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{8}$ Conductor: $$896$$$$\medspace = 2^{7} \cdot 7$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 8.0.1258815488.2 Galois orbit size: $2$ Smallest permutation container: $D_{8}$ Parity: odd Determinant: 1.56.2t1.b.a Projective image: $D_4$ Projective stem field: 4.0.1568.1

## Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 71 }$ to precision 6.

Roots:
 $r_{ 1 }$ $=$ $$7 + 38\cdot 71 + 45\cdot 71^{2} + 52\cdot 71^{3} + 50\cdot 71^{4} + 30\cdot 71^{5} +O(71^{6})$$ $r_{ 2 }$ $=$ $$14 + 27\cdot 71 + 58\cdot 71^{2} + 42\cdot 71^{3} + 46\cdot 71^{4} + 34\cdot 71^{5} +O(71^{6})$$ $r_{ 3 }$ $=$ $$27 + 16\cdot 71 + 17\cdot 71^{2} + 67\cdot 71^{3} + 31\cdot 71^{4} + 68\cdot 71^{5} +O(71^{6})$$ $r_{ 4 }$ $=$ $$33 + 67\cdot 71 + 6\cdot 71^{2} + 2\cdot 71^{3} + 5\cdot 71^{4} + 51\cdot 71^{5} +O(71^{6})$$ $r_{ 5 }$ $=$ $$38 + 3\cdot 71 + 64\cdot 71^{2} + 68\cdot 71^{3} + 65\cdot 71^{4} + 19\cdot 71^{5} +O(71^{6})$$ $r_{ 6 }$ $=$ $$44 + 54\cdot 71 + 53\cdot 71^{2} + 3\cdot 71^{3} + 39\cdot 71^{4} + 2\cdot 71^{5} +O(71^{6})$$ $r_{ 7 }$ $=$ $$57 + 43\cdot 71 + 12\cdot 71^{2} + 28\cdot 71^{3} + 24\cdot 71^{4} + 36\cdot 71^{5} +O(71^{6})$$ $r_{ 8 }$ $=$ $$64 + 32\cdot 71 + 25\cdot 71^{2} + 18\cdot 71^{3} + 20\cdot 71^{4} + 40\cdot 71^{5} +O(71^{6})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.