# Properties

 Label 2.83.3t2.a Dimension $2$ Group $S_3$ Conductor $83$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $$83$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 3.1.83.1 Galois orbit size: $1$ Smallest permutation container: $S_3$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.83.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$6 + 3\cdot 23 + 22\cdot 23^{2} + 18\cdot 23^{3} + 13\cdot 23^{4} +O(23^{5})$$ 6 + 3*23 + 22*23^2 + 18*23^3 + 13*23^4+O(23^5) $r_{ 2 }$ $=$ $$7 + 9\cdot 23 + 10\cdot 23^{2} + 11\cdot 23^{3} +O(23^{5})$$ 7 + 9*23 + 10*23^2 + 11*23^3+O(23^5) $r_{ 3 }$ $=$ $$11 + 10\cdot 23 + 13\cdot 23^{2} + 15\cdot 23^{3} + 8\cdot 23^{4} +O(23^{5})$$ 11 + 10*23 + 13*23^2 + 15*23^3 + 8*23^4+O(23^5)

### 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.