# Properties

 Label 2.31.3t2.b.a Dimension 2 Group $S_3$ Conductor $31$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $31$ Artin number field: Splitting field of $f= x^{3} + x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_3$ Parity: Odd Determinant: 1.31.2t1.a.a

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $12 + 47 + 33\cdot 47^{2} + 42\cdot 47^{3} + 14\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 2 }$ $=$ $13 + 21\cdot 47^{2} + 20\cdot 47^{3} + 19\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 3 }$ $=$ $22 + 45\cdot 47 + 39\cdot 47^{2} + 30\cdot 47^{3} + 12\cdot 47^{4} +O\left(47^{ 5 }\right)$

### 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.