# Properties

 Label 2.891.3t2.b Dimension $2$ Group $S_3$ Conductor $891$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $$891$$$$\medspace = 3^{4} \cdot 11$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 3.1.891.1 Galois orbit size: $1$ Smallest permutation container: $S_3$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.891.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$9 + 20\cdot 23 + 6\cdot 23^{2} + 7\cdot 23^{3} + 4\cdot 23^{4} +O(23^{5})$$ 9 + 20*23 + 6*23^2 + 7*23^3 + 4*23^4+O(23^5) $r_{ 2 }$ $=$ $$17 + 11\cdot 23 + 20\cdot 23^{2} + 12\cdot 23^{4} +O(23^{5})$$ 17 + 11*23 + 20*23^2 + 12*23^4+O(23^5) $r_{ 3 }$ $=$ $$20 + 13\cdot 23 + 18\cdot 23^{2} + 14\cdot 23^{3} + 6\cdot 23^{4} +O(23^{5})$$ 20 + 13*23 + 18*23^2 + 14*23^3 + 6*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.