# Properties

 Label 2.648.3t2.b Dimension $2$ Group $S_3$ Conductor $648$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $$648$$$$\medspace = 2^{3} \cdot 3^{4}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 3.1.648.1 Galois orbit size: $1$ Smallest permutation container: $S_3$ Parity: odd Projective image: $S_3$ Projective field: 3.1.648.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$15 + 78\cdot 83 + 19\cdot 83^{2} + 19\cdot 83^{3} + 78\cdot 83^{4} +O(83^{5})$$ $r_{ 2 }$ $=$ $$75 + 57\cdot 83 + 51\cdot 83^{2} + 26\cdot 83^{3} + 75\cdot 83^{4} +O(83^{5})$$ $r_{ 3 }$ $=$ $$76 + 29\cdot 83 + 11\cdot 83^{2} + 37\cdot 83^{3} + 12\cdot 83^{4} +O(83^{5})$$

### 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.