# Properties

 Label 2.896.8t6.a Dimension $2$ Group $D_{8}$ Conductor $896$ 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 number field: Galois closure of 8.0.1258815488.2 Galois orbit size: $2$ Smallest permutation container: $D_{8}$ Parity: odd Projective image: $D_4$ Projective field: 4.0.1568.1

## Galois action

### Roots of defining polynomial

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 values $c1$ $c2$ $1$ $1$ $()$ $2$ $2$ $1$ $2$ $(1,8)(2,7)(3,6)(4,5)$ $-2$ $-2$ $4$ $2$ $(1,2)(3,4)(5,6)(7,8)$ $0$ $0$ $4$ $2$ $(1,5)(2,7)(4,8)$ $0$ $0$ $2$ $4$ $(1,4,8,5)(2,6,7,3)$ $0$ $0$ $2$ $8$ $(1,7,4,3,8,2,5,6)$ $-\zeta_{8}^{3} + \zeta_{8}$ $\zeta_{8}^{3} - \zeta_{8}$ $2$ $8$ $(1,3,5,7,8,6,4,2)$ $\zeta_{8}^{3} - \zeta_{8}$ $-\zeta_{8}^{3} + \zeta_{8}$
The blue line marks the conjugacy class containing complex conjugation.