# Properties

 Label 2.3072.8t6.b.a Dimension $2$ Group $D_{8}$ Conductor $3072$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{8}$ Conductor: $$3072$$$$\medspace = 2^{10} \cdot 3$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 8.0.57982058496.8 Galois orbit size: $2$ Smallest permutation container: $D_{8}$ Parity: odd Determinant: 1.3.2t1.a.a Projective image: $D_4$ Projective stem field: 4.0.6144.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} + 24 x^{4} + 32 x^{2} + 24$$  .

The roots of $f$ are computed in $\Q_{ 283 }$ to precision 6.

Roots:
 $r_{ 1 }$ $=$ $$34 + 219\cdot 283 + 78\cdot 283^{2} + 8\cdot 283^{3} + 118\cdot 283^{4} + 108\cdot 283^{5} +O(283^{6})$$ $r_{ 2 }$ $=$ $$64 + 206\cdot 283 + 105\cdot 283^{2} + 262\cdot 283^{3} + 131\cdot 283^{4} + 197\cdot 283^{5} +O(283^{6})$$ $r_{ 3 }$ $=$ $$113 + 270\cdot 283 + 98\cdot 283^{2} + 224\cdot 283^{3} + 231\cdot 283^{4} + 123\cdot 283^{5} +O(283^{6})$$ $r_{ 4 }$ $=$ $$137 + 279\cdot 283 + 214\cdot 283^{2} + 66\cdot 283^{3} + 191\cdot 283^{4} + 210\cdot 283^{5} +O(283^{6})$$ $r_{ 5 }$ $=$ $$146 + 3\cdot 283 + 68\cdot 283^{2} + 216\cdot 283^{3} + 91\cdot 283^{4} + 72\cdot 283^{5} +O(283^{6})$$ $r_{ 6 }$ $=$ $$170 + 12\cdot 283 + 184\cdot 283^{2} + 58\cdot 283^{3} + 51\cdot 283^{4} + 159\cdot 283^{5} +O(283^{6})$$ $r_{ 7 }$ $=$ $$219 + 76\cdot 283 + 177\cdot 283^{2} + 20\cdot 283^{3} + 151\cdot 283^{4} + 85\cdot 283^{5} +O(283^{6})$$ $r_{ 8 }$ $=$ $$249 + 63\cdot 283 + 204\cdot 283^{2} + 274\cdot 283^{3} + 164\cdot 283^{4} + 174\cdot 283^{5} +O(283^{6})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.