# Properties

 Label 2.11025.8t5.a Dimension 2 Group $Q_8$ Conductor $3^{2} \cdot 5^{2} \cdot 7^{2}$ Frobenius-Schur indicator -1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8$ Conductor: $11025= 3^{2} \cdot 5^{2} \cdot 7^{2}$ Artin number field: Splitting field of $f= x^{8} - x^{7} - 34 x^{6} + 29 x^{5} + 361 x^{4} - 305 x^{3} - 1090 x^{2} + 1345 x - 395$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $Q_8$ Parity: Even Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{5}, \sqrt{21})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $42\cdot 79 + 27\cdot 79^{2} + 30\cdot 79^{3} + 73\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 2 }$ $=$ $7 + 63\cdot 79 + 39\cdot 79^{2} + 76\cdot 79^{3} + 39\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 3 }$ $=$ $25 + 29\cdot 79 + 38\cdot 79^{2} + 5\cdot 79^{3} + 26\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 4 }$ $=$ $45 + 41\cdot 79 + 21\cdot 79^{2} + 49\cdot 79^{3} + 25\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 5 }$ $=$ $49 + 16\cdot 79 + 55\cdot 79^{2} + 15\cdot 79^{3} + 37\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 6 }$ $=$ $53 + 79 + 12\cdot 79^{2} + 68\cdot 79^{3} + 15\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 7 }$ $=$ $66 + 72\cdot 79 + 42\cdot 79^{2} + 11\cdot 79^{3} + 57\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 8 }$ $=$ $72 + 48\cdot 79 + 78\cdot 79^{2} + 58\cdot 79^{3} + 40\cdot 79^{4} +O\left(79^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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