# Properties

 Label 3.229.4t5.a.a Dimension 3 Group $S_4$ Conductor $229$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $229$ Artin number field: Splitting field of 4.0.229.1 defined by $f= x^{4} - x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_4$ Parity: Even Determinant: 1.229.2t1.a.a Projective image: $S_4$ Projective field: Galois closure of 4.0.229.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 193 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $135 + 139\cdot 193 + 151\cdot 193^{2} + 188\cdot 193^{3} + 63\cdot 193^{4} +O\left(193^{ 5 }\right)$ $r_{ 2 }$ $=$ $145 + 66\cdot 193 + 184\cdot 193^{2} + 103\cdot 193^{3} + 30\cdot 193^{4} +O\left(193^{ 5 }\right)$ $r_{ 3 }$ $=$ $148 + 174\cdot 193 + 160\cdot 193^{2} + 29\cdot 193^{3} + 160\cdot 193^{4} +O\left(193^{ 5 }\right)$ $r_{ 4 }$ $=$ $151 + 4\cdot 193 + 82\cdot 193^{2} + 63\cdot 193^{3} + 131\cdot 193^{4} +O\left(193^{ 5 }\right)$

### 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 value $1$ $1$ $()$ $3$ $3$ $2$ $(1,2)(3,4)$ $-1$ $6$ $2$ $(1,2)$ $1$ $8$ $3$ $(1,2,3)$ $0$ $6$ $4$ $(1,2,3,4)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.