# Properties

 Label 2.1216.8t11.a Dimension $2$ Group $Q_8:C_2$ Conductor $1216$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8:C_2$ Conductor: $$1216$$$$\medspace = 2^{6} \cdot 19$$ Artin number field: Galois closure of 8.0.378535936.1 Galois orbit size: $2$ Smallest permutation container: $Q_8:C_2$ Parity: odd Projective image: $C_2^2$ Projective field: $$\Q(\sqrt{-2}, \sqrt{-19})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$30 + 54\cdot 137 + 7\cdot 137^{2} + 122\cdot 137^{3} + 33\cdot 137^{4} +O(137^{5})$$ $r_{ 2 }$ $=$ $$55 + 86\cdot 137 + 97\cdot 137^{2} + 55\cdot 137^{3} + 35\cdot 137^{4} +O(137^{5})$$ $r_{ 3 }$ $=$ $$71 + 34\cdot 137 + 42\cdot 137^{2} + 87\cdot 137^{3} + 72\cdot 137^{4} +O(137^{5})$$ $r_{ 4 }$ $=$ $$86 + 3\cdot 137 + 105\cdot 137^{2} + 126\cdot 137^{3} + 73\cdot 137^{4} +O(137^{5})$$ $r_{ 5 }$ $=$ $$89 + 44\cdot 137 + 119\cdot 137^{2} + 74\cdot 137^{3} + 93\cdot 137^{4} +O(137^{5})$$ $r_{ 6 }$ $=$ $$103 + 27\cdot 137 + 87\cdot 137^{2} + 77\cdot 137^{3} + 114\cdot 137^{4} +O(137^{5})$$ $r_{ 7 }$ $=$ $$120 + 98\cdot 137 + 126\cdot 137^{2} + 8\cdot 137^{3} + 132\cdot 137^{4} +O(137^{5})$$ $r_{ 8 }$ $=$ $$135 + 60\cdot 137 + 99\cdot 137^{2} + 131\cdot 137^{3} + 128\cdot 137^{4} +O(137^{5})$$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $2$ $2$ $1$ $2$ $(1,3)(2,7)(4,5)(6,8)$ $-2$ $-2$ $2$ $2$ $(1,6)(2,4)(3,8)(5,7)$ $0$ $0$ $2$ $2$ $(1,3)(4,5)$ $0$ $0$ $2$ $2$ $(1,2)(3,7)(4,8)(5,6)$ $0$ $0$ $1$ $4$ $(1,5,3,4)(2,6,7,8)$ $-2 \zeta_{4}$ $2 \zeta_{4}$ $1$ $4$ $(1,4,3,5)(2,8,7,6)$ $2 \zeta_{4}$ $-2 \zeta_{4}$ $2$ $4$ $(1,7,3,2)(4,6,5,8)$ $0$ $0$ $2$ $4$ $(1,6,3,8)(2,5,7,4)$ $0$ $0$ $2$ $4$ $(1,4,3,5)(2,6,7,8)$ $0$ $0$
The blue line marks the conjugacy class containing complex conjugation.