# Properties

 Label 8.38806720086016.36t555.b Dimension 8 Group $A_6$ Conductor $2^{18} \cdot 23^{6}$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $8$ Group: $A_6$ Conductor: $38806720086016= 2^{18} \cdot 23^{6}$ Artin number field: Splitting field of $f= x^{6} - x^{5} + x^{4} + 4 x^{3} - 6 x^{2} + 2 x + 2$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $A_6$ Parity: Even Projective image: $A_6$ Projective field: Galois closure of 6.2.71639296.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 641 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $210 + 349\cdot 641 + 367\cdot 641^{2} + 225\cdot 641^{3} + 591\cdot 641^{4} +O\left(641^{ 5 }\right)$ $r_{ 2 }$ $=$ $292 + 506\cdot 641 + 51\cdot 641^{2} + 40\cdot 641^{3} + 75\cdot 641^{4} +O\left(641^{ 5 }\right)$ $r_{ 3 }$ $=$ $367 + 327\cdot 641 + 583\cdot 641^{2} + 387\cdot 641^{3} + 364\cdot 641^{4} +O\left(641^{ 5 }\right)$ $r_{ 4 }$ $=$ $493 + 420\cdot 641 + 417\cdot 641^{2} + 521\cdot 641^{3} + 72\cdot 641^{4} +O\left(641^{ 5 }\right)$ $r_{ 5 }$ $=$ $599 + 165\cdot 641 + 550\cdot 641^{2} + 558\cdot 641^{3} + 188\cdot 641^{4} +O\left(641^{ 5 }\right)$ $r_{ 6 }$ $=$ $604 + 152\cdot 641 + 593\cdot 641^{2} + 188\cdot 641^{3} + 630\cdot 641^{4} +O\left(641^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $8$ $8$ $45$ $2$ $(1,2)(3,4)$ $0$ $0$ $40$ $3$ $(1,2,3)(4,5,6)$ $-1$ $-1$ $40$ $3$ $(1,2,3)$ $-1$ $-1$ $90$ $4$ $(1,2,3,4)(5,6)$ $0$ $0$ $72$ $5$ $(1,2,3,4,5)$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$ $72$ $5$ $(1,3,4,5,2)$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
The blue line marks the conjugacy class containing complex conjugation.