# Properties

 Label 4.5e2_11_29.8t35.4c1 Dimension 4 Group $C_2 \wr C_2\wr C_2$ Conductor $5^{2} \cdot 11 \cdot 29$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_2 \wr C_2\wr C_2$ Conductor: $7975= 5^{2} \cdot 11 \cdot 29$ Artin number field: Splitting field of $f=x^{8} - 2 x^{7} + 4 x^{6} - 5 x^{5} + 5 x^{4} - 5 x^{3} + 4 x^{2} - 2 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_2 \wr C_2\wr C_2$ Parity: Odd Determinant: 1.11_29.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 929 }$ to precision 6.
Roots: \begin{aligned} r_{ 1 } &= -149976192415541806 +O\left(929^{ 6 }\right) \\ r_{ 2 } &= -78517384993180892 +O\left(929^{ 6 }\right) \\ r_{ 3 } &= -179400370137705934 +O\left(929^{ 6 }\right) \\ r_{ 4 } &= -118739206366129749 +O\left(929^{ 6 }\right) \\ r_{ 5 } &= -194711489019800431 +O\left(929^{ 6 }\right) \\ r_{ 6 } &= -279769572953102002 +O\left(929^{ 6 }\right) \\ r_{ 7 } &= -308991004696435370 +O\left(929^{ 6 }\right) \\ r_{ 8 } &= 24450704703540344 +O\left(929^{ 6 }\right) \\ \end{aligned}

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $4$ $1$ $2$ $(1,4)(2,7)(3,5)(6,8)$ $-4$ $2$ $2$ $(2,7)(6,8)$ $0$ $4$ $2$ $(2,7)$ $2$ $4$ $2$ $(1,3)(2,6)(4,5)(7,8)$ $0$ $4$ $2$ $(2,7)(3,5)$ $0$ $4$ $2$ $(2,6)(7,8)$ $2$ $4$ $2$ $(1,4)(2,7)(6,8)$ $-2$ $4$ $2$ $(1,3)(2,7)(4,5)(6,8)$ $-2$ $8$ $2$ $(1,6)(2,3)(4,8)(5,7)$ $0$ $8$ $2$ $(1,4)(2,6)(7,8)$ $0$ $4$ $4$ $(1,3,4,5)(2,8,7,6)$ $0$ $4$ $4$ $(2,8,7,6)$ $2$ $4$ $4$ $(1,5,4,3)(2,7)(6,8)$ $-2$ $8$ $4$ $(1,6,4,8)(2,5,7,3)$ $0$ $8$ $4$ $(1,4)(2,8,7,6)$ $0$ $8$ $4$ $(1,3)(2,8,7,6)(4,5)$ $0$ $16$ $4$ $(1,6,3,2)(4,8,5,7)$ $0$ $16$ $4$ $(1,6)(2,5,7,3)(4,8)$ $0$ $16$ $8$ $(1,6,3,2,4,8,5,7)$ $0$
The blue line marks the conjugacy class containing complex conjugation.