# Properties

 Label 2.59.3t2.a Dimension $2$ Group $S_3$ Conductor $59$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $$59$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 3.1.59.1 Galois orbit size: $1$ Smallest permutation container: $S_3$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.59.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $8 + 15\cdot 17 + 15\cdot 17^{2} + 2\cdot 17^{3} + 12\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 2 }$ $=$ $12 + 15\cdot 17 + 5\cdot 17^{2} + 15\cdot 17^{3} + 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 3 }$ $=$ $14 + 2\cdot 17 + 12\cdot 17^{2} + 15\cdot 17^{3} + 2\cdot 17^{4} +O\left(17^{ 5 }\right)$

### 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 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.