# Properties

 Label 2.278784.8t5.d Dimension $2$ Group $Q_8$ Conductor $278784$ Indicator $-1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8$ Conductor: $$278784$$$$\medspace = 2^{8} \cdot 3^{2} \cdot 11^{2}$$ Frobenius-Schur indicator: $-1$ Root number: $1$ Artin number field: Galois closure of 8.0.5416809268248576.2 Galois orbit size: $1$ Smallest permutation container: $Q_8$ Parity: even Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{2}, \sqrt{33})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 13.
Roots:
 $r_{ 1 }$ $=$ $8 + 30\cdot 41 + 7\cdot 41^{2} + 10\cdot 41^{3} + 37\cdot 41^{4} + 21\cdot 41^{5} + 29\cdot 41^{6} + 4\cdot 41^{7} + 28\cdot 41^{8} + 37\cdot 41^{9} + 31\cdot 41^{10} + 21\cdot 41^{11} + 2\cdot 41^{12} +O\left(41^{ 13 }\right)$ $r_{ 2 }$ $=$ $11 + 15\cdot 41 + 22\cdot 41^{2} + 38\cdot 41^{3} + 36\cdot 41^{4} + 7\cdot 41^{5} + 35\cdot 41^{6} + 32\cdot 41^{7} + 41^{8} + 28\cdot 41^{9} + 39\cdot 41^{10} + 29\cdot 41^{11} + 20\cdot 41^{12} +O\left(41^{ 13 }\right)$ $r_{ 3 }$ $=$ $12 + 9\cdot 41 + 13\cdot 41^{2} + 32\cdot 41^{3} + 4\cdot 41^{4} + 13\cdot 41^{5} + 7\cdot 41^{6} + 22\cdot 41^{7} + 30\cdot 41^{8} + 13\cdot 41^{9} + 5\cdot 41^{10} + 5\cdot 41^{11} + 32\cdot 41^{12} +O\left(41^{ 13 }\right)$ $r_{ 4 }$ $=$ $20 + 22\cdot 41 + 21\cdot 41^{2} + 28\cdot 41^{3} + 25\cdot 41^{4} + 6\cdot 41^{5} + 41^{6} + 6\cdot 41^{7} + 10\cdot 41^{8} + 24\cdot 41^{9} + 16\cdot 41^{10} + 28\cdot 41^{11} + 27\cdot 41^{12} +O\left(41^{ 13 }\right)$ $r_{ 5 }$ $=$ $21 + 18\cdot 41 + 19\cdot 41^{2} + 12\cdot 41^{3} + 15\cdot 41^{4} + 34\cdot 41^{5} + 39\cdot 41^{6} + 34\cdot 41^{7} + 30\cdot 41^{8} + 16\cdot 41^{9} + 24\cdot 41^{10} + 12\cdot 41^{11} + 13\cdot 41^{12} +O\left(41^{ 13 }\right)$ $r_{ 6 }$ $=$ $29 + 31\cdot 41 + 27\cdot 41^{2} + 8\cdot 41^{3} + 36\cdot 41^{4} + 27\cdot 41^{5} + 33\cdot 41^{6} + 18\cdot 41^{7} + 10\cdot 41^{8} + 27\cdot 41^{9} + 35\cdot 41^{10} + 35\cdot 41^{11} + 8\cdot 41^{12} +O\left(41^{ 13 }\right)$ $r_{ 7 }$ $=$ $30 + 25\cdot 41 + 18\cdot 41^{2} + 2\cdot 41^{3} + 4\cdot 41^{4} + 33\cdot 41^{5} + 5\cdot 41^{6} + 8\cdot 41^{7} + 39\cdot 41^{8} + 12\cdot 41^{9} + 41^{10} + 11\cdot 41^{11} + 20\cdot 41^{12} +O\left(41^{ 13 }\right)$ $r_{ 8 }$ $=$ $33 + 10\cdot 41 + 33\cdot 41^{2} + 30\cdot 41^{3} + 3\cdot 41^{4} + 19\cdot 41^{5} + 11\cdot 41^{6} + 36\cdot 41^{7} + 12\cdot 41^{8} + 3\cdot 41^{9} + 9\cdot 41^{10} + 19\cdot 41^{11} + 38\cdot 41^{12} +O\left(41^{ 13 }\right)$

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

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

### Character values on conjugacy classes

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