# Properties

 Label 2.588.3t2.a Dimension $2$ Group $S_3$ Conductor $588$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $$588$$$$\medspace = 2^{2} \cdot 3 \cdot 7^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 3.1.588.1 Galois orbit size: $1$ Smallest permutation container: $S_3$ Parity: odd Projective image: $S_3$ Projective field: 3.1.588.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$33 + 19\cdot 61 + 35\cdot 61^{2} + 7\cdot 61^{3} + 3\cdot 61^{4} +O(61^{5})$$ $r_{ 2 }$ $=$ $$38 + 57\cdot 61 + 61^{2} + 34\cdot 61^{3} + 4\cdot 61^{4} +O(61^{5})$$ $r_{ 3 }$ $=$ $$52 + 44\cdot 61 + 23\cdot 61^{2} + 19\cdot 61^{3} + 53\cdot 61^{4} +O(61^{5})$$

### 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.