# Properties

 Label 2.3072.8t6.a.b 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: 8.0.57982058496.7 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 $\Q_{ 67 }$ to precision 7.

Roots:
 $r_{ 1 }$ $=$ $$2 + 19\cdot 67 + 46\cdot 67^{2} + 64\cdot 67^{3} + 49\cdot 67^{4} + 21\cdot 67^{5} + 61\cdot 67^{6} +O(67^{7})$$ $r_{ 2 }$ $=$ $$3 + 41\cdot 67 + 42\cdot 67^{2} + 3\cdot 67^{3} + 44\cdot 67^{4} + 46\cdot 67^{5} + 29\cdot 67^{6} +O(67^{7})$$ $r_{ 3 }$ $=$ $$10 + 34\cdot 67 + 42\cdot 67^{2} + 2\cdot 67^{3} + 62\cdot 67^{4} + 11\cdot 67^{5} + 14\cdot 67^{6} +O(67^{7})$$ $r_{ 4 }$ $=$ $$17 + 61\cdot 67 + 48\cdot 67^{3} + 31\cdot 67^{4} + 46\cdot 67^{5} + 15\cdot 67^{6} +O(67^{7})$$ $r_{ 5 }$ $=$ $$50 + 5\cdot 67 + 66\cdot 67^{2} + 18\cdot 67^{3} + 35\cdot 67^{4} + 20\cdot 67^{5} + 51\cdot 67^{6} +O(67^{7})$$ $r_{ 6 }$ $=$ $$57 + 32\cdot 67 + 24\cdot 67^{2} + 64\cdot 67^{3} + 4\cdot 67^{4} + 55\cdot 67^{5} + 52\cdot 67^{6} +O(67^{7})$$ $r_{ 7 }$ $=$ $$64 + 25\cdot 67 + 24\cdot 67^{2} + 63\cdot 67^{3} + 22\cdot 67^{4} + 20\cdot 67^{5} + 37\cdot 67^{6} +O(67^{7})$$ $r_{ 8 }$ $=$ $$65 + 47\cdot 67 + 20\cdot 67^{2} + 2\cdot 67^{3} + 17\cdot 67^{4} + 45\cdot 67^{5} + 5\cdot 67^{6} +O(67^{7})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.