# Properties

 Label 2.108.3t2.b Dimension 2 Group $S_3$ Conductor $2^{2} \cdot 3^{3}$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $108= 2^{2} \cdot 3^{3}$ Artin number field: Splitting field of $f= x^{3} - 2$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_3$ Parity: Odd Projective image: $S_3$ Projective field: Galois closure of 3.1.108.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $4 + 9\cdot 31 + 4\cdot 31^{2} + 8\cdot 31^{3} + 23\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 2 }$ $=$ $7 + 13\cdot 31 + 29\cdot 31^{2} + 11\cdot 31^{3} + 25\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 3 }$ $=$ $20 + 8\cdot 31 + 28\cdot 31^{2} + 10\cdot 31^{3} + 13\cdot 31^{4} +O\left(31^{ 5 }\right)$

### 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 values $c1$ $1$ $1$ $()$ $2$ $3$ $2$ $(1,2)$ $0$ $2$ $3$ $(1,2,3)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.