# Properties

 Label 2.42025.8t5.a.a Dimension 2 Group $Q_8$ Conductor $5^{2} \cdot 41^{2}$ Root number -1 Frobenius-Schur indicator -1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8$ Conductor: $42025= 5^{2} \cdot 41^{2}$ Artin number field: Splitting field of 8.8.74220378765625.1 defined by $f= x^{8} - 3 x^{7} - 63 x^{6} + 90 x^{5} + 1311 x^{4} - 20 x^{3} - 7702 x^{2} - 5524 x - 1009$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $Q_8$ Parity: Even Determinant: 1.1.1t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{5}, \sqrt{41})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 251 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $31 + 114\cdot 251^{2} + 222\cdot 251^{3} + 73\cdot 251^{4} +O\left(251^{ 5 }\right)$ $r_{ 2 }$ $=$ $56 + 6\cdot 251 + 134\cdot 251^{2} + 68\cdot 251^{3} + 186\cdot 251^{4} +O\left(251^{ 5 }\right)$ $r_{ 3 }$ $=$ $58 + 181\cdot 251 + 246\cdot 251^{2} + 82\cdot 251^{3} + 47\cdot 251^{4} +O\left(251^{ 5 }\right)$ $r_{ 4 }$ $=$ $82 + 29\cdot 251 + 179\cdot 251^{2} + 198\cdot 251^{3} + 231\cdot 251^{4} +O\left(251^{ 5 }\right)$ $r_{ 5 }$ $=$ $135 + 142\cdot 251 + 64\cdot 251^{2} + 167\cdot 251^{3} + 156\cdot 251^{4} +O\left(251^{ 5 }\right)$ $r_{ 6 }$ $=$ $174 + 30\cdot 251 + 117\cdot 251^{2} + 189\cdot 251^{3} + 73\cdot 251^{4} +O\left(251^{ 5 }\right)$ $r_{ 7 }$ $=$ $225 + 239\cdot 251 + 212\cdot 251^{2} + 77\cdot 251^{3} + 239\cdot 251^{4} +O\left(251^{ 5 }\right)$ $r_{ 8 }$ $=$ $246 + 122\cdot 251 + 186\cdot 251^{2} + 247\cdot 251^{3} + 245\cdot 251^{4} +O\left(251^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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