# Properties

 Label 4.5e3_11e2_29e2.8t29.3c1 Dimension 4 Group $(((C_4 \times C_2): C_2):C_2):C_2$ Conductor $5^{3} \cdot 11^{2} \cdot 29^{2}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $(((C_4 \times C_2): C_2):C_2):C_2$ Conductor: $12720125= 5^{3} \cdot 11^{2} \cdot 29^{2}$ Artin number field: Splitting field of $f=x^{8} - x^{7} + 3 x^{6} - 8 x^{5} + 9 x^{4} - 8 x^{3} + 16 x^{2} - 20 x + 9$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $(((C_4 \times C_2): C_2):C_2):C_2$ Parity: Even Determinant: 1.5.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 929 }$ to precision 7.
Roots: \begin{aligned} r_{ 1 } &= 160591522267966665177 +O\left(929^{ 7 }\right) \\ r_{ 2 } &= 179361339923196886811 +O\left(929^{ 7 }\right) \\ r_{ 3 } &= 26508495031979191456 +O\left(929^{ 7 }\right) \\ r_{ 4 } &= 160861733878600930251 +O\left(929^{ 7 }\right) \\ r_{ 5 } &= -145398331870544416033 +O\left(929^{ 7 }\right) \\ r_{ 6 } &= 163474160924526753914 +O\left(929^{ 7 }\right) \\ r_{ 7 } &= 283743415898996497344 +O\left(929^{ 7 }\right) \\ r_{ 8 } &= -231955813429226220310 +O\left(929^{ 7 }\right) \\ \end{aligned}

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

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

### Character values on conjugacy classes

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