# Properties

 Label 2.3e5.3t2.1c1 Dimension 2 Group $S_3$ Conductor $3^{5}$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $243= 3^{5}$ Artin number field: Splitting field of $f= x^{3} - 3$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_3$ Parity: Odd Determinant: 1.3.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots: \begin{aligned} r_{ 1 } &= 4 + 47\cdot 61 + 43\cdot 61^{2} + 4\cdot 61^{3} + 39\cdot 61^{4} +O\left(61^{ 5 }\right) \\ r_{ 2 } &= 5 + 26\cdot 61 + 49\cdot 61^{2} + 20\cdot 61^{3} + 32\cdot 61^{4} +O\left(61^{ 5 }\right) \\ r_{ 3 } &= 52 + 48\cdot 61 + 28\cdot 61^{2} + 35\cdot 61^{3} + 50\cdot 61^{4} +O\left(61^{ 5 }\right) \\ \end{aligned}

### 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.