# Properties

 Label 2.229.3t2.a.a Dimension 2 Group $S_3$ Conductor $229$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $229$ Artin number field: Splitting field of 3.3.229.1 defined by $f= x^{3} - 4 x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_3$ Parity: Even Determinant: 1.229.2t1.a.a Projective image: $S_3$ Projective field: Galois closure of 3.3.229.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $21 + 2\cdot 37 + 14\cdot 37^{2} + 35\cdot 37^{3} + 18\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 2 }$ $=$ $24 + 5\cdot 37 + 3\cdot 37^{2} + 5\cdot 37^{3} +O\left(37^{ 5 }\right)$ $r_{ 3 }$ $=$ $29 + 28\cdot 37 + 19\cdot 37^{2} + 33\cdot 37^{3} + 17\cdot 37^{4} +O\left(37^{ 5 }\right)$

### Generators of the action on the roots $r_{ 1 }, r_{ 2 }, r_{ 3 }$

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

### Character values on conjugacy classes

 Size Order Action on $r_{ 1 }, r_{ 2 }, r_{ 3 }$ Character value $1$ $1$ $()$ $2$ $3$ $2$ $(1,2)$ $0$ $2$ $3$ $(1,2,3)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.