# Properties

 Label 3.257e2.6t8.1 Dimension 3 Group $S_4$ Conductor $257^{2}$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $66049= 257^{2}$ Artin number field: Splitting field of $f= x^{4} + x^{2} - x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_4$ Parity: Even

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $73 + 47\cdot 157 + 11\cdot 157^{2} + 57\cdot 157^{3} + 146\cdot 157^{4} +O\left(157^{ 5 }\right)$ $r_{ 2 }$ $=$ $75 + 34\cdot 157 + 88\cdot 157^{2} + 46\cdot 157^{3} + 3\cdot 157^{4} +O\left(157^{ 5 }\right)$ $r_{ 3 }$ $=$ $77 + 138\cdot 157 + 50\cdot 157^{2} + 98\cdot 157^{3} + 122\cdot 157^{4} +O\left(157^{ 5 }\right)$ $r_{ 4 }$ $=$ $89 + 93\cdot 157 + 6\cdot 157^{2} + 112\cdot 157^{3} + 41\cdot 157^{4} +O\left(157^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

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

### Character values on conjugacy classes

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