# Properties

 Label 2.3332.8t11.c Dimension $2$ Group $Q_8:C_2$ Conductor $3332$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8:C_2$ Conductor: $$3332$$$$\medspace = 2^{2} \cdot 7^{2} \cdot 17$$ Artin number field: Galois closure of 8.0.8704143616.1 Galois orbit size: $2$ Smallest permutation container: $Q_8:C_2$ Parity: odd Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(i, \sqrt{119})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$20 + 22\cdot 53^{2} + 23\cdot 53^{3} + 26\cdot 53^{4} +O(53^{5})$$ 20 + 22*53^2 + 23*53^3 + 26*53^4+O(53^5) $r_{ 2 }$ $=$ $$24 + 11\cdot 53 + 48\cdot 53^{2} + 27\cdot 53^{3} + 14\cdot 53^{4} +O(53^{5})$$ 24 + 11*53 + 48*53^2 + 27*53^3 + 14*53^4+O(53^5) $r_{ 3 }$ $=$ $$27 + 29\cdot 53 + 9\cdot 53^{2} + 3\cdot 53^{3} + 26\cdot 53^{4} +O(53^{5})$$ 27 + 29*53 + 9*53^2 + 3*53^3 + 26*53^4+O(53^5) $r_{ 4 }$ $=$ $$32 + 32\cdot 53 + 16\cdot 53^{2} + 29\cdot 53^{3} + 35\cdot 53^{4} +O(53^{5})$$ 32 + 32*53 + 16*53^2 + 29*53^3 + 35*53^4+O(53^5) $r_{ 5 }$ $=$ $$37 + 11\cdot 53 + 26\cdot 53^{2} + 10\cdot 53^{3} + 27\cdot 53^{4} +O(53^{5})$$ 37 + 11*53 + 26*53^2 + 10*53^3 + 27*53^4+O(53^5) $r_{ 6 }$ $=$ $$40 + 50\cdot 53 + 16\cdot 53^{2} + 49\cdot 53^{3} + 24\cdot 53^{4} +O(53^{5})$$ 40 + 50*53 + 16*53^2 + 49*53^3 + 24*53^4+O(53^5) $r_{ 7 }$ $=$ $$42 + 9\cdot 53 + 12\cdot 53^{2} + 7\cdot 53^{3} + 41\cdot 53^{4} +O(53^{5})$$ 42 + 9*53 + 12*53^2 + 7*53^3 + 41*53^4+O(53^5) $r_{ 8 }$ $=$ $$45 + 12\cdot 53 + 7\cdot 53^{2} + 8\cdot 53^{3} + 16\cdot 53^{4} +O(53^{5})$$ 45 + 12*53 + 7*53^2 + 8*53^3 + 16*53^4+O(53^5)

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

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

### Character values on conjugacy classes

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