# Properties

 Label 3.2e4_83e2.6t8.2 Dimension 3 Group $S_4$ Conductor $2^{4} \cdot 83^{2}$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $110224= 2^{4} \cdot 83^{2}$ Artin number field: Splitting field of $f= x^{4} - 3 x^{2} - 2 x + 1$ 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 $\Q_{ 227 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $66 + 4\cdot 227 + 212\cdot 227^{2} + 44\cdot 227^{3} + 20\cdot 227^{4} +O\left(227^{ 5 }\right)$ $r_{ 2 }$ $=$ $82 + 109\cdot 227 + 218\cdot 227^{2} + 176\cdot 227^{3} + 97\cdot 227^{4} +O\left(227^{ 5 }\right)$ $r_{ 3 }$ $=$ $90 + 59\cdot 227 + 215\cdot 227^{2} + 190\cdot 227^{3} + 130\cdot 227^{4} +O\left(227^{ 5 }\right)$ $r_{ 4 }$ $=$ $216 + 53\cdot 227 + 35\cdot 227^{2} + 41\cdot 227^{3} + 205\cdot 227^{4} +O\left(227^{ 5 }\right)$

### 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.