# Properties

 Label 2.3_331.3t2.1c1 Dimension 2 Group $S_3$ Conductor $3 \cdot 331$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $993= 3 \cdot 331$ Artin number field: Splitting field of $f= x^{3} - x^{2} - 6 x + 3$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_3$ Parity: Even Determinant: 1.3_331.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots: \begin{aligned} r_{ 1 } &= 8 + 26\cdot 31 + 10\cdot 31^{2} + 23\cdot 31^{3} + 8\cdot 31^{4} +O\left(31^{ 5 }\right) \\ r_{ 2 } &= 11 + 31 + 28\cdot 31^{2} + 24\cdot 31^{3} + 16\cdot 31^{4} +O\left(31^{ 5 }\right) \\ r_{ 3 } &= 13 + 3\cdot 31 + 23\cdot 31^{2} + 13\cdot 31^{3} + 5\cdot 31^{4} +O\left(31^{ 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.