# Properties

 Label 2.712.8t6.b.b Dimension $2$ Group $D_{8}$ Conductor $712$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{8}$ Conductor: $$712$$$$\medspace = 2^{3} \cdot 89$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 8.0.2887553024.1 Galois orbit size: $2$ Smallest permutation container: $D_{8}$ Parity: odd Determinant: 1.712.2t1.b.a Projective image: $D_4$ Projective stem field: 4.0.5696.2

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 4 x^{6} + 13 x^{4} - 24 x^{3} + 20 x^{2} - 12 x + 9$$  .

The roots of $f$ are computed in $\Q_{ 179 }$ to precision 6.

Roots:
 $r_{ 1 }$ $=$ $$30 + 6\cdot 179 + 42\cdot 179^{2} + 96\cdot 179^{3} + 64\cdot 179^{4} + 63\cdot 179^{5} +O(179^{6})$$ $r_{ 2 }$ $=$ $$31 + 61\cdot 179 + 138\cdot 179^{2} + 101\cdot 179^{3} + 101\cdot 179^{4} + 121\cdot 179^{5} +O(179^{6})$$ $r_{ 3 }$ $=$ $$113 + 27\cdot 179 + 17\cdot 179^{2} + 128\cdot 179^{3} + 62\cdot 179^{4} + 131\cdot 179^{5} +O(179^{6})$$ $r_{ 4 }$ $=$ $$116 + 46\cdot 179 + 4\cdot 179^{2} + 111\cdot 179^{3} + 99\cdot 179^{4} + 31\cdot 179^{5} +O(179^{6})$$ $r_{ 5 }$ $=$ $$126 + 96\cdot 179 + 49\cdot 179^{2} + 159\cdot 179^{3} + 171\cdot 179^{4} + 113\cdot 179^{5} +O(179^{6})$$ $r_{ 6 }$ $=$ $$156 + 38\cdot 179 + 147\cdot 179^{2} + 32\cdot 179^{3} + 32\cdot 179^{4} + 66\cdot 179^{5} +O(179^{6})$$ $r_{ 7 }$ $=$ $$159 + 103\cdot 179 + 145\cdot 179^{2} + 111\cdot 179^{3} + 177\cdot 179^{4} + 133\cdot 179^{5} +O(179^{6})$$ $r_{ 8 }$ $=$ $$164 + 155\cdot 179 + 171\cdot 179^{2} + 153\cdot 179^{3} + 5\cdot 179^{4} + 54\cdot 179^{5} +O(179^{6})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.