# Properties

 Label 2.23.3t2.1c1 Dimension 2 Group $S_3$ Conductor $23$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $23$ Artin number field: Splitting field of $f= x^{3} - x^{2} + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_3$ Parity: Odd Determinant: 1.23.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $7 + 39\cdot 59 + 39\cdot 59^{2} + 36\cdot 59^{3} + 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 2 }$ $=$ $9 + 59 + 51\cdot 59^{2} + 39\cdot 59^{3} + 51\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 3 }$ $=$ $44 + 18\cdot 59 + 27\cdot 59^{2} + 41\cdot 59^{3} + 5\cdot 59^{4} +O\left(59^{ 5 }\right)$

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