# Properties

 Label 2.127449.8t5.b Dimension 2 Group $Q_8$ Conductor $3^{2} \cdot 7^{2} \cdot 17^{2}$ Frobenius-Schur indicator -1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8$ Conductor: $127449= 3^{2} \cdot 7^{2} \cdot 17^{2}$ Artin number field: Splitting field of $f= x^{8} - x^{7} + 65 x^{6} - 439 x^{5} + 1876 x^{4} - 12191 x^{3} + 60887 x^{2} - 124718 x + 121291$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $Q_8$ Parity: Even Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{17}, \sqrt{21})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $17 + 59\cdot 89 + 69\cdot 89^{2} + 71\cdot 89^{3} + 27\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 2 }$ $=$ $19 + 75\cdot 89 + 57\cdot 89^{2} + 78\cdot 89^{3} + 11\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 3 }$ $=$ $24 + 65\cdot 89 + 57\cdot 89^{2} + 13\cdot 89^{3} + 44\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 4 }$ $=$ $29 + 66\cdot 89 + 82\cdot 89^{2} + 11\cdot 89^{3} + 10\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 5 }$ $=$ $33 + 24\cdot 89 + 11\cdot 89^{2} + 31\cdot 89^{3} + 12\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 6 }$ $=$ $67 + 82\cdot 89 + 53\cdot 89^{2} + 29\cdot 89^{3} + 20\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 7 }$ $=$ $82 + 66\cdot 89 + 31\cdot 89^{2} + 46\cdot 89^{3} + 19\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 8 }$ $=$ $86 + 4\cdot 89 + 80\cdot 89^{2} + 72\cdot 89^{3} + 31\cdot 89^{4} +O\left(89^{ 5 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character values $c1$ $1$ $1$ $()$ $2$ $1$ $2$ $(1,4)(2,3)(5,8)(6,7)$ $-2$ $2$ $4$ $(1,3,4,2)(5,7,8,6)$ $0$ $2$ $4$ $(1,6,4,7)(2,5,3,8)$ $0$ $2$ $4$ $(1,8,4,5)(2,6,3,7)$ $0$
The blue line marks the conjugacy class containing complex conjugation.