# Properties

 Label 4.41e3_257e3.8t44.1c1 Dimension 4 Group $C_2 \wr S_4$ Conductor $41^{3} \cdot 257^{3}$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $4$ Group: $C_2 \wr S_4$ Conductor: $1169905924153= 41^{3} \cdot 257^{3}$ Artin number field: Splitting field of $f=x^{8} - x^{7} - x^{6} + x^{5} + x^{4} - x^{3} - x^{2} + x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_2 \wr S_4$ Parity: Even Determinant: 1.41_257.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 23.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $x^{2} + 45 x + 5$
Roots: \begin{aligned} r_{ 1 } &= 4125732070849904116955137095394880923 a + 108535691143585843174946962314492440731 +O\left(47^{ 23 }\right) \\ r_{ 2 } &= -4125732070849904116955137095394880923 a - 77122252044659842088034206978277200804 +O\left(47^{ 23 }\right) \\ r_{ 3 } &= 138459048631165464992074476883657365752 a + 61817149605088419169667164777055829502 +O\left(47^{ 23 }\right) \\ r_{ 4 } &= -138459048631165464992074476883657365752 a - 136719279473980099836979068573881549855 +O\left(47^{ 23 }\right) \\ r_{ 5 } &= 112425752280088062286582164550103345723 a - 52712332848855451861854777879647583767 +O\left(47^{ 23 }\right) \\ r_{ 6 } &= -112425752280088062286582164550103345723 a + 58518036824362378644838011434539641112 +O\left(47^{ 23 }\right) \\ r_{ 7 } &= 70856290574570332719634952915386126707 +O\left(47^{ 23 }\right) \\ r_{ 8 } &= -33173303780111579922219038009667703625 +O\left(47^{ 23 }\right) \\ \end{aligned}

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $4$ $1$ $2$ $(1,6)(2,5)(3,4)(7,8)$ $-4$ $4$ $2$ $(3,4)$ $-2$ $4$ $2$ $(2,5)(3,4)(7,8)$ $2$ $6$ $2$ $(1,6)(3,4)$ $0$ $12$ $2$ $(1,3)(2,7)(4,6)(5,8)$ $0$ $12$ $2$ $(1,2)(5,6)$ $-2$ $12$ $2$ $(1,7)(2,5)(3,4)(6,8)$ $2$ $24$ $2$ $(1,2)(3,4)(5,6)$ $0$ $32$ $3$ $(1,3,7)(4,8,6)$ $1$ $12$ $4$ $(1,3,6,4)(2,7,5,8)$ $0$ $12$ $4$ $(1,2,6,5)$ $2$ $12$ $4$ $(1,6)(2,5)(3,8,4,7)$ $-2$ $24$ $4$ $(1,3,6,4)(2,7)(5,8)$ $0$ $24$ $4$ $(1,2,6,5)(3,4)$ $0$ $48$ $4$ $(1,2,3,7)(4,8,6,5)$ $0$ $32$ $6$ $(2,7,3,5,8,4)$ $-1$ $32$ $6$ $(1,3,7)(2,5)(4,8,6)$ $1$ $32$ $6$ $(1,3,8,6,4,7)(2,5)$ $-1$ $48$ $8$ $(1,7,3,5,6,8,4,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.