# Properties

 Label 2.5e2_61e2.8t5.1c1 Dimension 2 Group $Q_8$ Conductor $5^{2} \cdot 61^{2}$ Root number -1 Frobenius-Schur indicator -1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8$ Conductor: $93025= 5^{2} \cdot 61^{2}$ Artin number field: Splitting field of $f= x^{8} - 3 x^{7} + 57 x^{6} + 135 x^{5} + 306 x^{4} + 5365 x^{3} + 23383 x^{2} + 40951 x + 75421$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $Q_8$ Parity: Even Determinant: 1.1.1t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 199 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $106\cdot 199 + 78\cdot 199^{2} + 61\cdot 199^{3} + 63\cdot 199^{4} +O\left(199^{ 5 }\right)$ $r_{ 2 }$ $=$ $3 + 25\cdot 199 + 148\cdot 199^{2} + 111\cdot 199^{3} + 58\cdot 199^{4} +O\left(199^{ 5 }\right)$ $r_{ 3 }$ $=$ $18 + 179\cdot 199 + 8\cdot 199^{2} + 14\cdot 199^{3} + 187\cdot 199^{4} +O\left(199^{ 5 }\right)$ $r_{ 4 }$ $=$ $30 + 41\cdot 199 + 63\cdot 199^{2} + 50\cdot 199^{3} + 23\cdot 199^{4} +O\left(199^{ 5 }\right)$ $r_{ 5 }$ $=$ $79 + 131\cdot 199 + 171\cdot 199^{2} + 154\cdot 199^{3} + 177\cdot 199^{4} +O\left(199^{ 5 }\right)$ $r_{ 6 }$ $=$ $118 + 37\cdot 199 + 58\cdot 199^{2} + 80\cdot 199^{3} + 74\cdot 199^{4} +O\left(199^{ 5 }\right)$ $r_{ 7 }$ $=$ $157 + 72\cdot 199 + 156\cdot 199^{2} + 13\cdot 199^{3} + 121\cdot 199^{4} +O\left(199^{ 5 }\right)$ $r_{ 8 }$ $=$ $195 + 3\cdot 199 + 111\cdot 199^{2} + 110\cdot 199^{3} + 90\cdot 199^{4} +O\left(199^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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