# Properties

 Label 2.120.8t11.c Dimension $2$ Group $Q_8:C_2$ Conductor $120$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8:C_2$ Conductor: $$120$$$$\medspace = 2^{3} \cdot 3 \cdot 5$$ Artin number field: Galois closure of 8.0.3240000.1 Galois orbit size: $2$ Smallest permutation container: $Q_8:C_2$ Parity: odd Projective image: $C_2^2$ Projective field: $$\Q(\sqrt{-6}, \sqrt{10})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$7 + 19\cdot 79 + 44\cdot 79^{2} + 40\cdot 79^{3} + 47\cdot 79^{4} +O(79^{5})$$ $r_{ 2 }$ $=$ $$17 + 74\cdot 79 + 66\cdot 79^{2} + 61\cdot 79^{3} + 63\cdot 79^{4} +O(79^{5})$$ $r_{ 3 }$ $=$ $$19 + 78\cdot 79 + 12\cdot 79^{2} + 15\cdot 79^{3} + 56\cdot 79^{4} +O(79^{5})$$ $r_{ 4 }$ $=$ $$32 + 66\cdot 79 + 21\cdot 79^{2} + 70\cdot 79^{3} + 68\cdot 79^{4} +O(79^{5})$$ $r_{ 5 }$ $=$ $$33 + 44\cdot 79 + 78\cdot 79^{2} + 65\cdot 79^{3} + 37\cdot 79^{4} +O(79^{5})$$ $r_{ 6 }$ $=$ $$38 + 72\cdot 79 + 68\cdot 79^{2} + 13\cdot 79^{3} + 12\cdot 79^{4} +O(79^{5})$$ $r_{ 7 }$ $=$ $$40 + 45\cdot 79 + 63\cdot 79^{2} + 67\cdot 79^{3} + 17\cdot 79^{4} +O(79^{5})$$ $r_{ 8 }$ $=$ $$54 + 73\cdot 79 + 37\cdot 79^{2} + 59\cdot 79^{3} + 11\cdot 79^{4} +O(79^{5})$$

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

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

### Character values on conjugacy classes

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