# Properties

 Label 2.119025.8t5.b Dimension $2$ Group $Q_8$ Conductor $119025$ Indicator $-1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8$ Conductor: $$119025$$$$\medspace = 3^{2} \cdot 5^{2} \cdot 23^{2}$$ Frobenius-Schur indicator: $-1$ Root number: $-1$ Artin number field: Galois closure of 8.0.1686221298140625.1 Galois orbit size: $1$ Smallest permutation container: $Q_8$ Parity: even Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{5}, \sqrt{69})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $5 + 7\cdot 139 + 19\cdot 139^{2} + 56\cdot 139^{3} + 131\cdot 139^{4} +O\left(139^{ 5 }\right)$ $r_{ 2 }$ $=$ $19 + 73\cdot 139 + 25\cdot 139^{2} + 93\cdot 139^{3} + 78\cdot 139^{4} +O\left(139^{ 5 }\right)$ $r_{ 3 }$ $=$ $30 + 53\cdot 139 + 93\cdot 139^{3} + 93\cdot 139^{4} +O\left(139^{ 5 }\right)$ $r_{ 4 }$ $=$ $69 + 30\cdot 139 + 101\cdot 139^{2} + 94\cdot 139^{3} + 2\cdot 139^{4} +O\left(139^{ 5 }\right)$ $r_{ 5 }$ $=$ $95 + 87\cdot 139 + 76\cdot 139^{2} + 116\cdot 139^{3} + 56\cdot 139^{4} +O\left(139^{ 5 }\right)$ $r_{ 6 }$ $=$ $101 + 19\cdot 139 + 7\cdot 139^{2} + 85\cdot 139^{3} + 113\cdot 139^{4} +O\left(139^{ 5 }\right)$ $r_{ 7 }$ $=$ $117 + 131\cdot 139 + 83\cdot 139^{2} + 53\cdot 139^{3} + 10\cdot 139^{4} +O\left(139^{ 5 }\right)$ $r_{ 8 }$ $=$ $123 + 13\cdot 139 + 103\cdot 139^{2} + 102\cdot 139^{3} + 68\cdot 139^{4} +O\left(139^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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